I would stay in Portsmouth personally...and then drive South on Rt 1 and go to North Hampton or Rye beaches.
I'm not a fan of Hampton in general though.
I would go to York Maine beaches
Hampton Beach recently recieved a 5 star rating for clean beaches. http://www.hamptonchamber.com/
and if you have kids in tow, they will like the area better than other area beaches. Hampton might be pricier than Portsmouth but its only a 15 minute ride down the coast, and a beautiful scenic ride also.
A couple of notes. That scenic ride down to Hampton is Rte 1A. Along that road, soon after it reaches the ocean, you will come to Wallis Sands State Park, and a couple of other state parks, all with very nice beaches and less crazy than Hampton.
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