In classic profile?? No, you cant change the opacity of the multimedia section there due to a bug.
Gaia is at fault there because they don't set the wmode in the HTML for the flash object.
I once researched this problem a year ago and one of the answer which I accepted and tested is this:
http://stackoverflow.com/questions/15161211/opacity-over-flash-doesnt-works
If I remember correctly, there also some other CSS properties which also does not work properly if wmode not set.
Considering the classic profile age, on 2003, setting the wmode increase the performance load which in terms of PC of that age is pretty significant, but as flash engine gets more updated of course the load is lighter and afterall today PC specs average is far better. However a different case for animated avatar since it is applied later on
If you check Youtube site, they also does not use wmode for the flash players used on the site to reduce performance load. If you want to est it out, use element inspector and just set the body to opacity of 0.1; . Pretty sure all that left over is only the flash. Except you have the HTML5 player option enabled on your YT account setting....