Are there any downsides to enabling Hardware Virtualization in the BIOS?

One reason is security.

Root kits could use it to gain higher access to hardware than OS. Setting it to disabled mitigates this risk (see analysis).

Performance-wise? No.

Security-wise? See @Josip's answer (which I've upvoted, btw)

However, common malwares can't yet leverage the virtualization extension. And if you use your computer safely (e.g., run as a regular user instead of an administrator-level one), the risks are practically mitigated.

That said, if you want to run virtual machines, Virtualization Extensions (Intel VT-x or AMD-V) will help increase performance significantly.