Opening of a business by an Indian citizen in USA
If you are already a citizen of the United States, opening your company in New York will be much easier (and not to mention cheaper) for you to do.
The first thing you should consider is whether an LLC or a Corporation would be right for you. An LLC would be more practical if you are a single or small two to three member partnership business; however in New York all new LLC's are required to go through publications, which depending on which county your business was registered can be a little on the expensive side. A corporation, on the other hand, does not require publications, but are subject to different tax responsibilities that may not be beneficial to your business. If you are a U.S. citizen, I would recommend exploring the possibility of forming an S-Corp, so you can avoid publications while being taxed as an LLC. I would advise consulting an accountant to find which entity would serve you and your business plan best.
If you are not a United States' citizen, you can still open your company in New York, there will just be a little more paperwork and a few extra fees, such as having your company's paperwork attached with an Apostille. This will allow you to open your company's bank account at home in India once is has been formed and registered in the state of New York.
I hope this answer was helpful.