Hastings is at its best in late spring and early summer, before the summer crowds arrive but when the weather is warm enough to enjoy the seafront. May is particularly good if you can time it with the Jack in the Green festival (May Bank Holiday weekend), which transforms the town.
The town is livelier than most English seaside resorts in autumn and winter — the independent shops, galleries, and restaurants that give Hastings its character don't close down in October as they might in a more resort-oriented town. The light on the East Sussex coast in autumn is exceptional, and the beach is emptier. Avoid school holidays if crowds bother you.