Google has failed me but it appears that it was perhaps the '96 or '00 Oly??? Also, did they start getting paid for WC medals before or after the first Oly with pay??? I suspect some swimmer on here knows the exact answers. 
'92 was the first Games that allowed professionals (cf. Dream Team). Countries and/or their NGBs/NSOs/NSFs do offer medal bonuses. The Olympics does not pay medal winners. Track & field was the first IF to offer prize money for Olympic medals beginning with these Games.
I wasn’t aware that swimming Olympic medallists were getting paid, outside of their own country’s Olympic committee. World Athletics started this Olympics to award gold medallists $50 000US, plus any prize money awarded by the athlete’s country. I remember reading that other athletes, outside of athletics, were jealous of this. That’s why I thought it was unique to athletics.
'92 was the first Games that allowed professionals (cf. Dream Team). Countries and/or their NGBs/NSOs/NSFs do offer medal bonuses. The Olympics does not pay medal winners. Track & field was the first IF to offer prize money for Olympic medals beginning with these Games.
I understand that the Olympics per se does not pay medal winners but rather it is USA Swimming and U.S. Olympic and Paralympic Committee (USOPC, formerly USOC) that pay the medalists. So you’re saying this started in '92??? I just don’t recall ever hearing that Mel Stewart and Mike Barrowman won any money with their golds in '92.
OK, so I went back and reviewed my Swimming World magazines from the '92, '96, '00, and '04 Oly, and I found no discussion whatsoever of $$$ for medals. It would appear that '08 was the first Oly where swimmers got paid for medals. In '08 it was 25K for gold, 15K for silver, and 10K for bronze.