It depends on the nature of the complaint, and the web host's policies in dealing with complaints.
Example of how I handle copyright complaints: When someone complains of one of my clients' copyright violations, they have to provide all information stipulated by the DMCA: they have to provide exact file names that are violating copyright, they can't just say "an entire site" is violating copyright. They also have to state that they (the complainer) are either the legal copyright holder, or are a representative for the legal copyright holder. (There are some other things they need to provide.)
Once that complaint is received, I notify the account owner, tell them that I'm going to temporarily remove certain files from their webspace because of a complaint, tell them which files are being removed, and why. The client then can respond to the person who filed the complaint stating that the files were not violating copyrights. Once they do that, they then either leave those files taken down or ask that they be restored after 14 days have elapsed from the date of the original complaint. If they ask for the files to be restored and have replied that they do not believe that the files in question violate copyright, those files are restored to a customer's webspace. Then, if the complaintant believes that the files *do* violate copyrights, they can then proceed with legal action against the account holder - whose contact information they will now have, from the rebuttal notice the client filed. Any further legal action is solely between the client and the complaintant, at that point.
If the client just leaves the files down and does not file a rebuttal notice...then those files are deleted from the "holding area" after 14 days, and the complaintant goes on their merry.
There are all kinds of other things that might happen: a client could re-upload the files, or upload them under new names. Since client content isn't monitored, if the copyright holder catches them displaying the copyrighted material without a rebuttal notice being sent, they can file a follow-up complaint. If this happens too many times, the client may be suspended or terminated (after we try speaking with them to figure out if they just don't understand the process, or if they're just scoffing at the received complaints.) A complaintant could file multiple complaints against the same "infringing" files, even after the client has filed their rebuttal notice. This hasn't happened; but if it does, I'll speak with a lawyer about my options in ignoring repeated complaints from the individual. And so on, and so forth.
That's just an example of how copyright complaints may be handled. Termination is never the first step unless the client has been installing and running scripts that compromise server or network security - and even then, suspension is preferred before complete termination, and shutting down the offending scripts is the preferred action before suspension. There are clearly defined escalations in dealing with problems, whether client-generated or external complaints; but none of them start with suspension, let alone termination. They all take into account keeping us from being held legally liable from our clients' actions, but at the same time give people a way to lodge honest complaints against our clients. They just won't be able to win a lawsuit against TLM Network for the content of our members' sites.
If a web host makes it easy for people to lodge complaints against their clients; if a web host does police egregious network abuses so that their servers don't generate more spam or host warez sites; if a web host is a "good neighbor" rather than a nanny; it won't be simple for someone to prove willful negligence or abetting lawbreaking activities. No guarantees, of course. But you'll make their work pretty dang difficult.