I'm afraid there aren't really any budget airlines that fly from Thailand to India -- budget carrier Tiger Airways does fly to India, but sadly not from Thailand. It does, however, fly from Singapore, so you might want to look into flying to Singapore from Thailand (pretty much all of the low-carrier airlines do some version of this route: check Air Asia and Jetstar Asia as well as Tiger) and then on to India. It's worth noting, also, that Tiger only flies into Bangalore, Chennai, and Kochi.
Otherwise, in order to get from Thailand to India you might have to graduate away from budget carriers and look into other mainstream airlines like Thai Airways, Air India, Sahara, or Jet. You'll have to determine for yourself whether it's cheaper (and worth it) to do a budget flight from Thailand to Singapore and then on to India, or whether it's worth just ponying up and paying for a non-budget carrier.
Good luck! Have a great trip.
Which airline you choose depends on destination. Thai fly from Bangkok to Delhi, Mumbai, Kolkata, Chennai, Bangalore and Hyderabad. Air India also fly to many of those plus you have Jet Airways also.
There is no budget airline as such on the routes. Druk Air also fly the Bangkok to Kolkata route