The formation of helium depended on the abundance of free neutrons, since neutrons will easily combine with protons to form deuterium, a required pre-requisite to forming helium. The ratio of free neutrons to free protons in the early universe just prior to what's called Big Bang Nucleosynthesis (BBN) was about 7 protons to ever 1 neutron. Once the temperature of the universe fell to the point that deuterium was stable, this opened up a shot time frame where the universe was cool enough for fused nuclei to remain stable and hot enough for fusion to occur.
Since the proton-neutron ratio was 7 to 1, combining neutrons with protons quickly gobbles up all the free neutrons to form deuterium, which leaves 6/7 parts of the matter as hydrogen (free protons) and 1/7 parts deuterium. This deuterium then quickly fuses to form either helium-3 or tritium, followed by another round of fusion that leads to Helium.
The key here lies in the fact that protons by themselves have a very, very difficult time fusing together to forum deuterium since proton-proton fusion depends on the very small cross section of the weak interaction. In the core of the Sun, each time two protons come together in a possible fusion event, there is only a very tiny chance of them actually fusing together. However, deuterium is much easier to fuse, so the formation of helium is highly dependent on how much deuterium is available, which itself depends on the amount of neutrons available. Without neutrons, there simply wouldn't have been enough time for proton-proton fusion to create much deuterium. (Consider that the Sun has been shining for 4+ billion years thanks to proton-proton fusion and has only converted a fraction of its hydrogen into helium, whereas Big Bang Nucleosynthesis only lasted for about 20 minutes)
So in the end it comes down to the original ratio of protons to neutrons. Had the ratio been much higher there would be much more helium created from BBN.
See here for more:
http://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis#Helium-4