To facilitate multimessenger studies with TeV and PeV astrophysical neutrinos, the IceCube Collaboration has developed a realtime alert system for the highest confidence and best localized neutrino events. In this work we investigate the likelihood of association between realtime high-energy neutrino alerts and explosive optical transients, with a focus on core-collapse supernovae (CC SNe) as candidate neutrino sources. We report results from triggered optical follow-up observations of two IceCube alerts, IC170922A and IC171106A, with Blanco/DECam ($gri$ to 24th magnitude in $\sim6$ epochs). Based on a suite of simulated supernova light curves, we develop and validate selection criteria for CC SNe exploding in coincidence with neutrino alerts. The DECam observations are sensitive to CC SNe at redshifts $z \lesssim 0.3$. At redshifts $z \lesssim 0.1$, our selection criteria reduce background SNe contamination to a level below the predicted signal. For the IC170922A (IC171106A) follow-up observations, we expect that 12.1% (9.5%) of coincident CC SNe at $z \lesssim 0.3$ are recovered, and that on average, 0.23 (0.07) unassociated SNe in the 90% containment regions also pass our selection criteria. We find two total candidate CC SNe that are temporally coincident with the neutrino alerts, but none in the 90% containment regions, which is statistically consistent with expected rates of background CC SNe for these observations. Given the signal efficiencies and background rates derived from this pilot study, we estimate that to determine whether CC SNe are the dominant contribution to the total TeV-PeV energy IceCube neutrino flux at the $3\sigma$ confidence level, DECam observations similar to those of this work would be needed for $\sim200$ neutrino alerts, though this number falls to $\sim60$ neutrino alerts if redshift information is available for all candidates.