I have a travertine stone sink, which has small pits in the side wall and base near the plug hole. We are finding that although regularly cleaned mould can form in these pits. Is there a way of coating the sink with a thin layer of clear resin that will fill in these pits and stop the mould from forming.

Why not use stuff formulated for the job?
There are various stone cleaners and sealers eg http://www.travertinedirect.co.uk/categories/Sealing-and-Cleaning/
I installed a bathroom at home with travertine sinks and tiles - sealed everything once and the sinks three times, nice natural low gloss finish and has worked just fine.

If the pitting is bad you might be able to use the stone cleaner and then localised repair with clear epoxy and then seal over the whole sink.  If going down that route probably as easy to remove the sink and do as Fred suggests although go at it very gently so as not to make the pitting worse.

Good luck
Stuart