Can I access the museum from Boston by train i.e. Public transportation?

Yes of course

You can take the commuter rail from Boston and hop into a city bus. Check the maps, or just take the 23, 26, 24, or 30 to Lincoln Square (pull it before the brutalist police station) and walk up Highland Street towards the museum, about 3 blocks up and 2 over.

We took the train out of South Station, then walked the 8-tenths of a mile from Union Station to the museum. Returned on the 5:20pm train.

Train to Worcester then Uber to the museum. But.... Watch the times for your return to Boston.... Trains are not too often.

Yes. From Boston, I took TRAIN, then a cab to museum. I was visiting from out of town.

Commuter Rail runs to Worcester

Yes