Of course it does, otherwise a malicious mirror can (theoretically) work to find a collision between their malware and the legitimate file and serve you the former.
There's no good reason not to use a secure hash function.
If your threat model involves an attacker who is able to achieve a hash collision while still implanting a sophisticated malware, you should probably avoid downloading software from random websites...
They have moved away from SHA-1, and are now using SHA-256. Previous releases were signed with SHA-1, and before that it was MD5: https://handbrake.fr/checksums.php
I'm not sure why they haven't retroactively calculated checksums for older versions.
These researchers have found the first "freestart" collision, and they estimate the SHA-1 collision cost to take a few months, costing between 75K$ and 120K$.
Practically speaking, I don't think anyone could make a profit by forging a Handbrake release, but the FBI probably do have some very high-profile targets who use video encoding software.