I think that depends on how you sign off on the job with your client. You should certainly hang on to a backup of the original 'as-is' database, but the decision about who does the regular backups would be by agreement. Your client should at least be made aware of the risks and the need to backup regularly, and either he has the means and the know-how to do it, or he pays you (or someone else) an ongoing amount to backup the data at agreed intervals.
I have been thinking about that too, and was meaning to put something on the wishlist for a simple client-manageable backup and scheduler program that could sit in a site admin area, and either set to automatic interval, or run at the push of a button.
I was looking at these: Auto backup and phpJobScheduler (which looks simpler that what I've read about cron jobs) and was wondering if WebAssist could create something similar but in a friendly GUI. I wouldn't use them on my site until someone who knows their stuff checked it out though.
I may play with them on a dispensable database, and see what happens.