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We model the evolution of the He\,I lines, especially He\,I $1.08\mu$m and He\,I $2.06\mu$m, from the envelope of SN 1987A during the first two years after the explosion. The He\,I metastable $2\,^3\!S$ and $2\,^1\!S$ states are populated by fast electrons resulting from energy deposition by gamma rays, and depopulated by electron collisions, two-photon decay, and Penning ionization of H\,I. The $1.08\mu$m and $2.06\mu$m lines are excited by thermal electron impact, recombination, and photoexcitation by a UV continuum. We can fit the observed evolution of He\,I $2.06\mu$m with a model in which the helium in the emitting $(v < 2500\, {\rm km\, s}^{-1})$ region has mass ${\rm M}_{\rm He} \sim 1.5 {\rm M}_\odot$ and contains $\sim 5\%$ of hydrogen. Only $\sim 30 \%$ of the observed $1.08\mu$m line comes from He\,I; this feature is actually dominated by [S\,I]$1.08\mu$m.
