Morning glories (Ipomoea spp.) are true annuals in most climates — they complete their life cycle in one growing season and die at the first hard frost. There's no "cutting down for winter" because winter kills them completely.
Ipomoea indica (Blue Dawn Flower) and I. learii are perennial in zones 9–11. In frost-free areas, they become woody vines needing hard pruning in late winter. But the classic garden morning glory (I. purpurea, I. tricolor) is annual everywhere with frost.
If you want morning glories in the same spot next year: do nothing. They'll volunteer enthusiastically. If you want them elsewhere: collect seeds, scatter in desired location after last frost.
No pruning shears required — just a rake and a decision on whether to encourage or discourage next year's volunteers.