We have had a bit a water problem in our basement. When it rains hard the rain rolls down the exterior basement steps pretty hard and the drain at the bottom of the steps is not big enough to keep up with the heavy rainfalls.

It is OK to install a trench drain at the stop of the stairs and connect it to a pipe and route the water that way? Not sure if using a trench this way is OK.

Thanks for the help in advance.

Sure you can do that. If you can get enough pitch to your French drain to gravity feed the water away. Use perferated rigid PVC or soft black plastic drain tubing that's fabric covered. Backfill with couarse gravel no fines.

Bob gave you good advice. Treat the french drain.(see image) just like any drainage and slope the trench away from the house. Good luck, Tom

Or install collection pit at the lowest point, put in sump-pump with automatic float actuator. Pump water away from the house.