Recently, the supercooled Wahnstrom binary Lennard-Jones mixture was partially crystallized into MgZn(2) phase crystals in lengthy molecular dynamics simulations. We present molecular dynamics simulations of a modified Kob-Andersen binary Lennard-Jones mixture that also crystallizes in lengthy simulations here, however, by forming pure fcc crystals of the majority component. The two findings motivate this paper that gives a general thermodynamic and kinetic treatment of the stability of supercooled binary mixtures, emphasizing the importance of negative mixing enthalpy whenever present. The theory is used to estimate the crystallization time in a Kob-Andersen mixture from the crystallization time in a series of related systems. At T=0.40 we estimate this time to be 5x10(7) time units ( approximately 0.1 ms). A new binary Lennard-Jones mixture is proposed that is not prone to crystallization and faster to simulate than the two standard binary Lennard-Jones mixtures. This is obtained by removing the like-particle attractions by switching to Weeks-Chandler-Andersen type potentials, while maintaining the unlike-particle attraction.