The Public Broadcasting Service (PBS) is an American network of broadcasters. Many of its affiliates chose to screen episodes of the 1963 version of Doctor Who. Doctor Who has never, however, been carried by the PBS network itself.
The heyday of PBS association with Doctor Who — that is the time when the number of stations carrying Doctor Who was not in decline — was roughly from the late 1970s to the mid 1980s. A substantial number of stations continued carrying the programme well into the 1990s, and a two or three still do so as of 2010. Tom Baker is likely the only version of the Doctor who has been screened on every PBS station that carried Doctor Who.
How Doctor Who came and left PBS Edit
Unlike most broadcast networks in the US that are funded either through commercial sales or cable fees, PBS is funded by government support, corporate grants, and donations from viewers. As such, it is able to broadcast without commercial interruption — other than its own continuity announcements and appeals for donations. However, the piecemeal nature of PBS' funding means that the network's hold over its individual affiliates is comparatively weak. Schedules vary widely from station to station, depending upon the resources of a particular station, and the desires of its local viewership. PBS station viewers can successfully lobby for their local station to carry a particular show, if they are able to bring together enough other viewers willing to pay for its purchase.
It is because of this dynamic that the 1963 version of Doctor Who found its way onto individual PBS stations in the 1970s, why it was able to remain there in most states through the early 1980s, and also why many PBS stations began to shed themselves of Doctor Who in the late 1980s and 1990s. The need for individuals willing to specifically earmark donations for Doctor Who also explains why the BBC Wales series has not been widely shown on PBS stations. PBS have been denied first-run rights by BBC Worldwide, in deference to true network debuts on Sci Fi and BBC America. By the time that Worldwide have made the show available to PBS stations, most American fans have already had a chance to see the episodes on network television or to buy the DVDs. Thus, a PBS broadcast is superfluous to most modern American viewers, and cannot generate the same level of financial support that it did in previous decades. A common rationale among modern fans is that a $100 donation to a local PBS station in the hope that they carry Doctor Who could simply go towards the DVD purchase of Doctor Who for the certainty of being able to view the episodes.
All that being said, some PBS stations do still air the program, including WTTW Channel 11 in Chicago. 
The PBS audience Edit
Whereas an NBC or CBS affiliate is owned for profit, such is not really the purpose of a PBS station. Therefore few have the ability to stand completely on their own, receiving only the donations of their viewers and federal grants. Most require a physical presence at or near universities, if only to get a financial discount on their physical plant. Many PBS stations so located also have a reduced overall labour cost than their completely independent counterparts, because they can easily get interns to work in the studios. Many PBS stations also choose to be located close to universities because much of their local documentary programming is easier to create if they are close to local scholars who provide the on-air talent.
The practical result of PBS affiliates being wed to universities is that the audience for Doctor Who in its American heyday was significantly student-based. PBS stations usually offered the strongest signals to campus televisions when most people still received television signals aerially. The effect of Doctor Who upon student populations in America was thus pronounced just because of the particular channel that carried the programme.
The general shape of PBS stations' buying patterns Edit
Although a handful of pioneering PBS stations may have begun broadcasting Doctor Who with a few Jon Pertwee stories (Chicago's WTTW showed some episodes in 1975, sometimes without a discernible order),  Doctor Who broadcasting on PBS stations largely began with the Time + Life distribution of seasons 12 to 15. These initially appeared as individual episodes, with Howard Da Silva linking narration at the top of episodes to explain things for viewers who may have forgotten where the narrative had left off. These three seasons were put onto a loop my most of the stations that carried them, because the stations had bought the right to show the episodes a certain number of times. The fact that this initial package ended with The Invasion of Time allows us to date general American involvement with Doctor Who to no earlier than about 1978.
PBS would then obtain rights to subsequent seasons. Usually, the prospect of getting the new season was used as an incentive during PBS donation requests, or "pledge drives". As each new season was added to the library of the particular station, the "Tom Baker loop" would grow ever longer. Each station would go as far as they could with their Baker episodes, then snap back to Robot, and do it all again. Thus, Tom Baker is particularly vivid in the minds of many American fans because his adventures were played so often. Eventually, the Peter Davison and Colin Baker seasons were added to many PBS station libraries.
Arguably, the influence of the PBS broadcasts reached their zenith on November 23, 1983, the 20th anniversary of Doctor Who, when PBS broadcasters aired the anniversary special The Five Doctors, several days before the BBC.
Sales to PBS stations began to drop off around the time The Trial of a Time Lord went up for sale. There were a variety of reasons. One of the biggest was the advent of the VHS recorder. This allowed American fans to copy episodes of Doctor Who, which were being replayed on a regular basis. It was possible for viewers in some markets to get a complete Tom Baker-Colin Baker run in about a year, if they were so inclined. Also, BBC Video had begun releasing official NTSC VHS editions of stories at that point. Thus, there was simply less actual need for a PBS station. Negative press surrounding the Colin Baker and Sylvester McCoy eras may have played a contributing factor, but this cannot be proven, and certainly many PBS stations continued to air the series all the way through to Survival, which began to appear on PBS broadcasters in Detroit and elsewhere in 1990.
It was around this time that many PBS stations chose to look backward instead of forwards. In the late 1980s, they began to more strongly buy existing serials from the William Hartnell, Patrick Troughton and Jon Pertwee eras.
The PBS "omnibus" Edit
PBS broadcasts of Doctor Who in the 1980s and 1990s often used an "omnibus" format. Instead of airing each individual episode on a weekly or daily basis, each serial was edited together into a movie format. This meant that individual broadcasts could be as short as 45 minutes (for broadcasts of 2-episode stories), to 90 minutes which was the most common length for airing a stiched-together four-episode story, to a four hour marathon in the case of The War Games (The Trial of a Time Lord, however, was never broadcast as a single omnibus, but was aired in four instalments broken down by the generally-defined story arcs — three 90-minute episodes followed by a 45-minute conclusion). Due to the independent nature of affiliate stations, however, viewers in one part of the United States often had a very different experience of Doctor Who than those in another. Some stations exclusively aired omnibus editions, some never did. And some stations broadcast omnibuses on the weekend, while transmitting standard episodes during the week.
The omnibus format varied in quality with regards to episode transitions. Sometimes cliffhangers were edited together smoothly, while other times the "electronic scream" episode-ending sound effect and sometimes a few moments of the episode opening credits were accidentally left in (this occurred frequently with Davison-era episodes). Some Pertwee omnibus episodes would switch from colour to black and white and back again in the case of serials where only B&W prints of some episodes survived; Invasion of the Dinosaurs was the only incomplete serial to be broadcast by PBS, with its then-missing first episode omitted (when the missing episode was recovered, a complete omnibus was compiled).
The omnibus style broadcasts were not universal; for example, California station KTEH would run episodes individually on weeknights, and as an omnibus late on one weekend night. PBS affiliates also produced their own documentaries based upon the series, utilising behind-the-scenes footage and interviews with cast and crew at American Doctor Who conventions. KTEH, notably, would bring some of the actors portraying the Doctor to their studio to film original interviews (as well as meet American fans).
PBS and BBC Wales Edit
When the Doctor Who resumed producation again in 2005, PBS did not initially receive broadcast rights to the newer episodes, which instead went to the Sci Fi Channel, and in 2009 the rights for first broadcast were taken over by BBC America. However, PBS affiliates subsequently received rebroadcast rights to Series 1 of Doctor Who starring Christopher Eccleston. These episodes aired in summer of 2007 and Series Two episodes aired in the summer of 2008. Individual PBS affiliates as of 2010 have been airing not only Series 4 episodes, but also instalments of Doctor Who Confidential.
On 22 November 1987, one PBS broadcaster of Doctor Who received international media attention when Horror of Fang Rock, being aired on WTTW in Chicago, became part of the infamous Max Headroom broadcast signal intrusion incident.