We present ultraviolet, optical, and infrared photometry of the afterglow of the X-ray flash XRF 050416A taken between approximately 100 s and 36 days after the burst. We find an intrinsic spectral slope between 1930 and 22200 Å of β=-1.14+/-0.20 and a decay rate of α=-0.86+/-0.15. There is no evidence for a change in the decay rate between approximately 0.7 and 4.7 days after the burst. Our data imply that there is no spectral break between the optical and X-ray bands between 0.7 and 4.7 days after the burst and are consistent with the cooling break being redward of the Ks band (22200 Å) at 0.7 days. The combined ultraviolet/optical/infrared spectral energy distribution shows no evidence for a significant amount of extinction in the host galaxy along the line of sight to XRF 050416A. Our data suggest that the extragalactic extinction along the line of sight to the burst is only approximately AV=0.2 mag, which is significantly less than the extinction expected from the hydrogen column density inferred from X-ray observations of XRF 050416A assuming a dust-to-gas ratio similar to what is found for the Milky Way. The observed extinction, however, is consistent with the dust-to-gas ratio seen in the Small Magellanic Cloud. We suggest that XRF 050416A may have a two-component jet similar to what has been proposed for GRB 030329. If this is the case, the lack of an observed jet break between 0.7 and 42 days is an illusion due to emission from the wide jet dominating the afterglow after approximately 1.5 days.