Why not contact Scarborough Tourist Office as they'll have lists of accommodation.
My holidays were with my parents 30-odd years ago but I remember enjoying the Norbreck and Cumberland Hotels ... but not the Red Lea because at that time it was a geriatric wonderland (believe it's been updated since!).
You could also try The Grand and any hotels on St Nicholas Cliff but they were too posh for me and mine! You'll have a great time though, and take your swimming cossie!