The immune system exploits a wide range of strategies to combine sensitivity with selectivity for optimal response. We propose a generic physical mechanism that allows tuning the location and steepness of the response threshold of cellular processes activated by multivalent binding. The mechanism is based on the possibility to modulate the attraction between membrane receptors. We use theory and simulations to show how tuning interreceptor attraction can enhance or suppress the binding of multivalent ligand-coated particles to surfaces. The changes in the interreceptor attraction less than the thermal energy kBT can selectively switch the receptor-clustering and activation on or off in an almost step-wise fashion, which we explain by near-critical receptor density fluctuations. We also show that the same mechanism can efficiently regulate the onset of endocytosis for, e.g., drug delivery vehicles.