in between not too far from the train . I don't know the size of Levanto so I don't know how far the center is or the beach.
Go to Booking.com and use the map feature to find lodgings in Levanto. You can see the train station and the beach on the map for reference.
In the past I have stayed at Hotel Garden (close to the beach, small rooms) and at Affittacamere Marco (a modest B&B in the town). Affittacamere Marco was a good budget option, Hotel Garden is more expensive. The walk to the train station is up a slight hill but still an easy walk from town.
On our recent Village Italy tour we stayed at the Albergo Primavera. It was downhill from the train station (so is everything else) and a short walk to the beach. Our room was fine, with a tiny balcony. The staff was great. I don't know the price range, because it was included in the tour, but I would definitely consider staying there again. Oh, there as a small outdoor covered area, where we could sit, talk, drink wine...