The exact requirements vary from state to state, but in New York it appears that there is no requirement for you to get a NY Drivers License (as opposed to some other states where once you have been a resident for more than 3 months your home country drivers license is no longer valid).
However on a J1 visa you can optionally get a local drivers license, as long as your visa length is greater than 1 year, AND you have at least 6 months of validity on the visa remaining.
As far as whether it's worth doing this for the purpose of insurance, it depends on what type of insurance you are after.
For rental car insurance, there is no difference in cost of the insurance depending on the type of license you have. If you are getting the insurance from the rental car company (CDP/LDW/etc) then they will not vary it depending on the type of license you have. If you are getting it via travel insurance or credit card insurance then again it will not vary based on your license - but be sure to read the fine print to make sure it's covered!
If you are looking at buying a car, then it will make a difference, but be prepared for a shock either way. Many insurance companies will not offer you insurance if you do not have a US drivers license. I've been told there are some that will - especially for medium-term tourists - but expect to pay a lot for it.
If you do have a US drivers license then you will be able to get coverage, but it will not be cheap. Every insurance company I've ever looked at will ask you how long you've been licensed in the US, and will penalize you heavily for anything below 18 months/3 years/5 years (depending on the company). Most will also ask how long you've held a license in another country, and may reduce the price slightly due to your overseas license - but it won't be by much.
Plugging your details, along with those of the type of car you might consider buying, into some of the insurance companies websites will probably be very telling for you - both in terms of whether they will allow you to purchase coverage without a US license, and how much they will charge with only 1-2 months license history in the US.