Botanical Gardens has its own Metro-North stop on the Hudson Line. There's nothing direct from where you are.

Best bet would be to get to one of the stations on the Harlem line by coming east and then just taking the train to and from.

Here's the map

Otherwise, you can come into Grand Central and go back and forth from there but I think the other way is more efficient.

Here is a link that might be useful: Metro-North Harlem Line

Have to chirp-in that the Botanical Gardens stop is on the Harlem Line, not the Hudson Line. (I think that's clear from your link, but wanted to clarify) I know that there is connecting bus-service across the Tappan-Zee between Orange County and the Hudson Line's Tarrytown Station. It is possible to take the Hudson line to 125th Street and change there for the stops north, as yet another alternative.

Enjoy your visit! It's lovely there.

Sorry, I meant Harlem line. I was looking also at Hudson (visible also on map). Botanical stop is on Harlem and they don't cross. But it should be easy to plan once you know where you want to get the train.

It's the most fabulous day and place. I always feel better after a trip to the Botanical gardens and the way they take care of everything is so instructive. The only problem is how much work it takes to get everything that way!!!

Another alternative: take Metro North from Harriman Sta. (ample parking) to Suffern stop at Chestnut and Ramapo, walk a couple of blocks to Chestnut and Hallett and get Tappan Zee Express bus to White Plains Transportation Ctr, where you catch Metro North (Harlem line) to Botanical Gardens.
I also endorse Sweetclg's suggestion.

thanks for your help, gave me a few ways to go. I wish there was a bus from Hairraman to the gardens. That would make it easier.. Now I'll see which way my sister wants to go.

Whichever way you get there it is so worth the trip. Their techniques are so amazing I took my camera and found myself photographing everything from the way the trees were mulched to the stone walkways! The food's not to bad in the caf either.