Amorphous polymers are among the most common materials used in adhesives, and a clear understanding of the effects of molecular scale features on macroscopic responses is necessary to design new, better performing adhesives. While many features have been investigated, including effects of molecular weight, inclusion of filler materials, and some effects of crosslinking, much of the understanding of the adhesive response remains empirical. Specifically, choosing the appropriate combination of polymer properties that optimize the work required to debond is still a challenge and the interplay between mechanical and chemical properties of polymers at interfaces is largely unknown. Here, we perform molecular dynamics simulations on a simple coarse-grained polymer model to directly investigate the role of crosslinking in determining the adhesive response of amorphous polymers at the molecular level. We find that crosslinking has a dramatic effect on the mechanical properties even at relatively low crosslink densities, and that crosslinking alone can be effective for optimizing the adhesive response of amorphous polymer adhesives. We observe a clear transition from cohesive to adhesive failure as the crosslink density is increased which coincides with the optimal toughening in the films. Furthermore, we find that our model captures the key molecular scale deformation mechanisms that control the adhesive response. For low crosslink densities, increased crosslinking improves the adhesive response by inhibiting chain sliding and allowing the structures to achieve large deformations, but as the crosslink density is increased further, the adhesive response is diminished due to reduced overall deformability. Our results provide simple but important insights into how crosslinking in amorphous polymer adhesives can be used to tune the mechanical response and ultimately to optimize adhesive performance for various applications.