It's hard to say without looking at the traffic myself, trying to analyze what exactly they're doing specifically. Are they just trying to scrape your website, or trying to run malicious scripts? Things like that can give you a bit to go on.
In the latter case, enabling WAF rules could help, but you have to take a bit of care with those to ensure you don't cause false-positive triggers.
Blocking ASN's of entire web hosts where the malicious traffic is coming from is probably not a bad idea. Even if it seems like a hopeless endeavor, it may just take some time, monitoring, and persistence.
To some extent, though, you do have to account for this kind of traffic occurring and be able to scale with it as your website grows.