Yes. You definitely should.
Use Let's Encrypt to get a free SSL certificate (or as many as you want or need). SSL is always good to have: without it, man-in-the-middle attacks will be launched, by the NSA if no-one else.
Most multiplayer games (e.g. Minecraft) just use raw packets. There's little reason to encrypt packets after one is authenticated as only the network service provider (e.g. Verizon/Virgin/TalkTalk) can fake TCP packets.
However, if you're hosting your game on your website then the whole thing has to be in SSL (unless you want to have a sign in page over SSL and another page with no SSL, cringe).
Advantages of SSL:
- With Let's Encrypt, the certificates are automatic and free
- Actual security
- A sense of security for users (browser status bar shows lock icon)
- Privacy, no NSA spying
- +1 PageRank in search engines for using SSL
- Negligible performance cost
- If your game has such high performance requirements that WebSocket SSL is a deal-breaker, then your game is probably too intensive for browsers to begin with, SSL or not.
- I can't think of any others.