Yes they will
I think that in some cases, having cheaper contraceptives will make them more affordable and therefore encourage their usage. It won't be true in every case but in a lot of instances it will make them more used. I think that birth control should be free or very cheap so that people will be more encouraged to use them. It is better that they are used than not, because people are going to participate in sexual activity whether or not they use protection.
It would be nice, but not encouraging
Making contraceptives more affordable would ease the burden for some, but I do not think it would increase the overall usage. Those committed to protection would not let the cost of contraceptives hinder their acquisition. There are programs, such as Planned Parenthood, that provide contraceptives for little or no cost to those that seek it.