There are trains from Springfield/Hartford/BDL to New Haven on Amtrak at a pretty decent frequency. You'll find BDL is near the Windsor Locks Amtrak station and for less than $20 you can catch a train right into New Haven Union Station. There is a CT Bus from BDL to the Amtrak Station, but I did an uber the last time I was up there and it was about $10.
More info about the bus:
https://www.ifly.com/bradley-interna...transportation