Nope. Silk is not plastic and silk is usually very soft and won't effect bettas fins.
Do you have a filter in your tank, as long finned bettas have somewhat delicate fins and the constant flow of a filter, even when it's baffled can still effect the fins.
How large is the tank
From constantly swimming long finned bettas delicate tailed can also become a bit shredded
What's the substrate
From experience I've noticed that bettas with gravel substrate in there tank have ragged fins