In theory all the major browsers support SVG and all but Microsoft's IE support SVG declarative animation. The quality of that support is variable and bugs come and go. In consequence, any statements made here may not be true tomorrow. The statements below refer to declarative animation in SVG

Chrome
Chrome support is quite good still but its performance is not as good as it was a few years ago.
Firefox
Until June 2015, it had comparable performance to Chrome but recently performance has degraded some what.
Safari
Performance is usually good on Apple products. Not really available under some other platforms.
Opera
Used to be as good as the opposition but has fallen away recently
IE
NO NATIVE SUPPORT AT ALL but there is/was a project FakeSmile which did quite a good job of providing declarative animation support in IE. It did not support all features but was reasonably comprehensive.
Edge
Good support for SVG declarative animation.

Overall, there seems to have been little attempt to compile declarative animation efficiently. As the state of the animation is known at compile time and will not change, unlike scripted animation, it should be possible to make a very efficient implementation that can handle large SVG animations. Unfortunately the main browsers do not seem particularly interested in providing that support.