In turn, would that increase our gravitational pull?
The Earth loses mass because hydrogen and helium (plus other elements in trace amounts compared to hydrogen and helium) escapes the Earth's atmosphere. The Earth gains mass because incoming asteroids (most of them very small) impact the Earth's atmosphere; a few make it all the way to the surface of the Earth. Whether those incoming asteroids burn up in the atmosphere or make it all the way to the surface is irrelevant; the Earth gains mass.
Whether the net result is a mass gain or mass loss is a bit up in the air; the uncertainties on both are rather large, and they overlap. The Earth might be gaining or losing a tiny bit of mass every year. That said, most research leans toward atmospheric losses being greater than mass gain from impacting asteroids, comets, and dust.
Whichever is the case, it's a bit irrelevant. Even the most extreme upper estimates on mass loss or mass accumulation are incredibly tiny compared to the mass of the Earth itself.
The Earth has a net loss of mass each year. The infall of debris from space is more than matched mainly by the loss of hydrogen from the atmosphere.
According to the BBC radio show More or Less the net annual loss of mass is ~50,000 tons/year.
Most of it burns up in the atmosphere and never actually makes it the surface. So no, as far as I know or have ever heard/read the Earth is not getting more massive. Any increase in mass due to space debris (or other things like more humans being born) is negligible compared to the mass of the Earth anyways so it has a null effect in terms of the gravitational attraction.