If you haven't already done so, define the minimum system requirements of your website, e.g.
- supported browsers¹,
- minimum display size,
- required cookie permissions,
If the user did not satisfy the minimum system requirements, it's not your fault that it didn't work for him. Investigate the issue, prove that the minimum system requirements were not satisfied and send the client an invoice for the time you spent doing this.²
Of course, in some cases it's not so easy: You might be convinced that the problem lies "on the user's side", but you might not be able to prove it without putting a lot of effort into it. In that case, you should talk to your customer:
I have performed some tests, and I'm pretty sure that the problem is a weird firewall configuration/a buggy IE plugin/etc. However, to prove this, I'll have to put a lot of effort into it. If I do that and it turns out that the fault was not on my side, I will have to send you a bill for the work done by me. Are you sure that you want me to continue investigating this issue?
¹ This doesn't mean it won't work with other browsers, it only limits your warranty to these browsers. Usually, the customer will understand that you cannot test your web site extensively with every browser out there. Ideally this should be cleared up-front:
Support for IE8-10, FF12-19 and Safari 5 is included in the offer. IE7 can be included for an additional $xxx, IE6 for an additional $xxxx.
² Sending an invoice is a very powerful sign: Even if the customer complains and you end up canceling the invoice as a goodwill gesture, the client learns that unjustified complaints can cost money.