Can you tell me are e-cigarettes safe to smoke around babies?

Yes and no: If an infant ever got access to one and ingested or sucked the nicotine out of it they could die of nicotine poisoning. This does remove the issue of toxic smoke.

Nobody knows: Because the companies that make e-cigarettes managed to wiggle past any kind of regulation by the fda, nobody really knows because there is no research. All e-cigarettes are not created equal, and while it is clear that the sidestream smoke is nowhere near as toxic as smoked tobacco, nobody can say whether it is safe. Don't smoke around babies, e-cigarettes or other. Consider complete cessation.