Really depends on how it happens.
If we assume a larger customer base in Europe - then how does this happen? Stronger (and wealthier) Roman Empire?
Or are China for some reason selling more at a drastically lower price? In which case, why?
If the former, then stronger Romans, with more silk, probably just means that there is a larger economy to draw on, making the Romans more likely to rebuild their empire.
If the latter? Whilst there would be lots of silk being sold, the security on producing the silk may fall - partially because peasants are more like to afford to make a living on selling silk at these lower prices, and aren't going to be able to afford security. Even if it is nobles, the increased quantity over quality means that the same costs for security have to stretch further. This makes it more likely that they'll slip up, and that silkworms may make it out of china in greater numbers - eventually allowing local silk producers to out-bid Chinese silk, requiring them to lower the price even more.
Persia, either way, gets a lot of silk tariffs - but probably less profit off of ensuring the trade routes are safe - because less tax per merchant, but also the increased traffic would likely mean more raiders trying to take advantage of the route.