Baseball in Japan wasn't banned through WW2. It was actually used as a morale booster much like sports had been used in the US. Something citizens and soldiers could do to attempt to take their minds off the war at hand. At the time it wasn't really popular and they only had one professional team in 1934. By the time WW2 rolled around there was a league called the Japanese Baseball League. From 1936 to 1950 there were 12 separate teams involved. This was the main baseball league in Japan at the time and it ran for every single year until 1950, except for 1945 when war escalated for the worse in Japan.

In 1950, the JBL reformed into the Nippon Professional Baseball league and it continues to be extremely popular in Japan to this day. Americans pushed hard in the aftermath of WW2 to make baseball a big thing in Japan and it succeeded.

EDIT for sources:

http://www.yamasa.org/acjs/network/english/newsletter/things_japanese_17.html

http://baseballguru.com/jalbright/japanseasons.html

http://www.baseballinwartime.com/japanese.htm

http://en.wikipedia.org/wiki/Japanese_Baseball_League

http://en.wikipedia.org/wiki/Nippon_Professional_Baseball#History