Though the claims sound promising, so far no one has proved that Himalayan salt lamps release negative ions, let alone enough to have any impact on health. Most of the research so far has used negative ions from other sources, not lamps.
In a couple of small studies on rats and mice, contact with a salt lamp had antidepressant and anti-anxiety effects. That doesn't mean the lamps would have the same effect on humans. Researchers would have to test the theory.
Though a small amount of pollutants in the air might stick to salt rock, these rocks don't have anywhere near the filtering ability of, say, charcoal, a common component of air filters.
A Himalayan salt lamp might bring a nice decorative touch and a warming glow to your space, but there's no research right now that says it will improve your health in a big way.