No, gene patents do not encourage biotechnological innovation. Indeed, they do the opposite and stifle innovation as once a particular company has a gene patent locked down, no other companies may attempt to see if they can improve upon that gene in any way. That is the very definition of stifling innovation.
Human beings, through their natural curiosity and desire to improve the life of themselves and others, have been innovating technologically for literally thousands of years. Patents are a modern form of protection given to inventors, bestowing to them exclusive right to their invention for years. That can actually stifle innovation, since others are not permitted to work on the patented invention (in this case a gene) and perhaps innovate without fear of a lawsuit.