Don't use the heater. It could cause a fire and it wouldn't do much for you. What type of stain are you using? An oil base stain could be applied down to 40 degrees as long as there is no moisture during and after application. I suspect moisture is the culprit.
Try to wait until the temps are better. One way to finish your deck is to wait until midday, allowing the deck to warm up a little and hopefully dry a little, then apply the stain. Stain complete boards, a few at a time. Try to allow a couple of hours of sunlight to help dry the stain. It could take a while doing it this way.