The phrase "to become stable" means to reach a state of balance or equilibrium, where conditions are no longer changing rapidly and have settled into a consistent pattern. Full definition
Dating an older man offers the opportunity of becoming stable in terms of finances. (petesgrillandpizza.com)
The good news is that those ups and downs will come to an end; like hot flashes, they should disappear once your hormones become stable again. (health.com)
In fact, however, the pool of potential marriage partners for single mothers in impoverished communities does not include many men with good prospects for becoming stable and helpful partners. (contemporaryfamilies.org)