What does 拜 mean in 拜在俄做生意的中国有钱人所赐，他们的入境手续都不合法?
拜X所賜, Y is a standard expression which means Y happens, thanks to X or if it weren't for X, Y wouldn't have happened. It may also carry all the sarcasm or irony contained in the English version. If you want to decompose the structure, it literally means acknowledging the bestowed/conferred upon X, this causes Y.

拜在俄做生意的中國有錢人所賜，他們的入境手續都不合法，所以一查就給錢，一查就給錢，一來二去警察都知道中國人好給錢了
Thanks to the wealthy Chinese people doing business in Russia, who enter the country without following proper procedures and underhandedly pay the inspection/police officers every time they get checked, all the police officers [who are inside this shady bribery business] now know [or more logically, are now under the impression] that Chinese people love giving bribes.