Can a decay tooth cause a large swollen node at the base of the throat?

Infection. Lymph nodes act as a trap for infection. If you have infected teeth and/or infected gums you may have resultant swollen lymph nodes. See hour dentist immediately for an evaluation. If you have no dental problems the next visit should be with your physician. The source of infection must be located and treated. Antibiotics will slow spread of infection, but will not "cure" a tooth infection.
Swollen Nodes. Any infection in the head, regardless of the cause, can drain into the lymph nodes causing swelling. You need to see at least a dentist, and i would also advise ENT specialist as soon as possible. We can't assume the swelling is either an inflamed\infected node or from a tooth infection. These situations can go downhill fast. If necessary go to er of a hospital for an evaluation and treatment!
Ent. Yes it can. It could indicate you have an abscessed tooth. Seeing a dentist promptly would be advisable. If not able to see dentist at least a round of antibiotics would be prudent until able to see dentist.