I used to fly DC-3's on a scheduled basis to STT, and yes, there can be tailwinds landing to the east, but generally not very strong...usually 5 knots or less.
However, there CAN
be significant turbulance/windshear while landing, if the wind is from the north, due to the rather high bluffs on the north side of the airport.
Up until the runway was extended, most airlines operating turbojet equipment classified STT as a 'Captains only' landing airfield.
Don't know about now.