Why didn't helium continue to generate after a few minutes post-Big Bang? Helium stopped generating after a few minutes after the Big Bang, but what made it impossible for its generation to continue?  
 A: To form He, one needs to go through intermediate deuterium synthesis.
This is only possible during a window of time when: (1) the universe is not too hot/dense that deuterium cannot survive for long enough to take part in further reactions before being destroyed by energetic photons; (2) the universe is not too cool to synthesise deuterium in the first place.
It is the latter constraint that ends primordial He synthesis, after about 5-10 minutes, though I guess the decay of free neutrons also makes a (very) small contribution.
Helium production does of course continue inside main sequence stars to the present day. The He content of the universe and the interstellar and intergalactic media are very slowly increasing as the products of stellar nucleosynthesis are disseminated by stellar winds and supernovae.
A: Helium formation is quite unlikely. Directly after the big bang, there were plenty of unbound particles - protons, neutrons and electrons. As the universe was incredibly tiny, the density was unimaginably high. In these conditions, two protons and a neutron could fuse to form He-3. As the universe grew, densities fell, temperatures dropped, and the chances of this fusion happening fell to essentially zero.
Later, stars would for. In these stars, gravity re-created fairly high densities and temperatures. Luckily, those were still far, far lower than at the Big Bang - else stars would burn through their hydrogen in minutes instead of billions of years.
A: For a reaction to be in equilibrium in the expanding Universe, the interaction rate $\Gamma$ has to be much bigger than the rate of expansion of the Universe, which is the hubble parameter $H$. At some point, $\Gamma$ dropped below $H$, and the interaction stopped occurring (it "froze-out", in cosmology jargon), and the abundance of Helium stopped changing. (Until stars formed, of course.)
