Due to heavy bombing in February 1945 and the use of tunnels as strategic military hideouts, much of Berlin's transit system was effectively inoperable by the end of the war (the U-Bahn dated to 1902 and the S-Bahn to the 1870s). Construction commenced after the war ended but it took until the early 1950s to be restored to its pre-war state. Here is a map of the U-Bahn system in 1952.

The construction of the Berlin Wall did not mark the beginning of the division of Berlin's public transit, as the S-Bahn had already been operated by the Soviet Military Administration immediately following the end of the war. After a number of disputes regarding whether East or West would serve as headquarters of the BVG, West Berlin's mayor, Ernst Reuter, agreed to split the BVG between East and West in April 1949, thereby forming BVG-West and BVG-East. The GDR was founded October 7th, 1949, and already owned a fair share of Berlin's now-divided (in terms of administration, at least) public transit. The U-Bahn continued to run freely between states until the construction of the Berlin Wall, but both bus and tram services ceased to run between East and West by 1952. West Berliners had to acquire passes to travel to East Berlin by 1952 as well.

East German border sealing in August 1961 caused the closure/bypassing of 16 stations and 12 lines. S-Bahn and U-Bahn lines that were run by BVG-West but were physically located on East German soil, or near the border (U6, U8, S1, and S2 lines) were at first secured, but then eventually became "ghost stations".

U- and S-Bahn ghost stations were staffed by border security and either bypassed by trains (as in trains would run through without stopping) or disused completely, depending on their location. For example, the now-busy Potsdamer Platz station was a ghost station from 1961 until 1990, and its S-Bahn line would bypass trains without stopping through the station as it veered through East Berlin. The U-Bahn line would simply turn back. Here's a BVG-West map from 1977 that documents ghost stations as well as functioning East lines. Most of these stations reopened around July 1990.

Although a little philosophical, this is a good academic source that runs through the history (with images), and it helped me map out the U-Bahn's changes as written above: "Identities in transit: the (re)connections and (re)brandings of Berlin's municipal railway infrastructure after 1989" by Samuel Merrill. Otherwise, there are plenty of non-academic sources and articles that can be easily found by Googling something like "U Bahn cold war". Also, if you're interested in photography, there are a number of pictures of what transit looked like in Berlin during the Cold War on photographer Chris John DeWitt's blog.