Jade Empire was Bioware trying to ride the mini craze of chinese fantasy cinema that got relatively popular in the early 00s with movies like Crouching Tiger, Hidden Dragon, Hero and House of the Flying Daggers.
Hero and House of Flying Daggers only came out in America in 2004, they didn’t have anything to do with Jade Empire getting made. Crouching Tiger, Hidden Dragon may have. But Jackie Chan, Jet Li, and other Hong Kong movies had been huge throughout the '90s (at least by the mid ‘90s) in the rental market. Like, Jackie Chan and Jet Li were already pretty well known when Rush Hour and Lethal Weapon 4 came out in ‘98. Turns out Martial Law started in ‘98 as well.
Either way, those didn't have the impact that Crouching Tiger did (in general, I don't know if anyone at Bioware ever actually mentioned specific inspirations). That movie was huge at the time and was being talked up everywhere.
In the OXM interview (that’s in the magazine) they did list off a few movies. I don’t really remember what they were, but Magnificent Butcher was one I think. Yes, Madam may have been one too. I don’t think any of the movies they listed were really in the same style of fantasy as what they’re doing in Jade Empire. Star Wars feels like it was about as big an influence on Jade Empire as it was their Star Wars game. There is a event, not quite a quest, where you come across a kid and can either save them or given them a sword to kill their attack which is basically the story of Casca meeting Griffith in Berserk...the ‘97 Berserk anime had been out for a few years before Jade Empire came out, so it’s possible they got that from there. I know I’d seen the Berserk anime a few years before Jade Empire came out anyways, and when that moment came up, I was like: Berserk.
As for impact, depends on what “those” are. The pre Crouching Tiger Hong Kong movies that were big from like ‘95 up to Crouching Tiger definitely had more impact than Crouching Tiger did. Those (combined with the Hong Kong handover) are why you had Hong Kong actors and directors primed for Hollywood. Crouching Tiger, Hidden Dragon could have been why BioWare thought it was a good time do Jade Empire; but I’d guess it came about from them renting kung fu movies in the mid ‘90s.
The Matrix, and how big a deal they made of Yuen Woo-ping doing the fight choreography for it, probably had a bigger impact than Crouching Tiger too.