‘to start’ and ‘to starting’, prefaced by ‘key’

Solution 1:

Both can be correct, but have different meanings. The key to these sentences is actually the gerund (~ing ending).

The start of something is an instantaneous moment in time. There is only one start, and once something has started, it does not start again.

The starting of something is a duration of time including the start, but possibly also the initial preparations, and also continuing for an indeterminate duration.

For an example, consider a foot race. The start of the race is when the gun goes off. But the starting of the race includes when the runners walked up to the start line, and might also include the early portion of the race itself where the runners are jockeying for initial position.

The key to start someone's communication is the key for the very instant of their initial contact.

Patience is the key to start their communication. (because they expect they will need to use that patience soon)

The key to starting someone's communication is the key for the early portion of their communication.

Patience is the key to starting their communication. (because as they are talking, they are frustrating each other)