Question

Why do alkali metals have low density and increases down the group

Answer 1

Why do alkali metals have low density

Alkali metals have very high reactivity, so the smaller the ionization energy count the more reactive it gets, Alkali metals have the smallest ionization energy being in Group 1 which is why they have low density. Even they (the alkali metals) should not be exposed to oxygen at all or they will easily react.
and also due to it's tiny cohesive energy which makes it soft and easy to cut. The smaller the cohesive energy the softer it gets.

Why do alkali metals increase down the group

Density=mass/volumeDensity=mass/volume
1) If mass is increasing and volume is decreasing, then density (mass/volume) will increase.
2)If mass is decreasing and volume is increasing simultaneously, then the density (mass/volume) will decrease.
3)If both mass and volume are increasing, then we need to check which one of them is increasing at a faster rate(since both are contradictory factors)
a) If mass is increasing at a faster rate than volume, then density will increase.
b) If volume is increasing at a faster rate than mass (i.e denominator in mass/volume is increasing making the overall fraction smaller), then density decreases.
You can apply same logic when both volume and mass are decreasing simultaneously.
Generally, we see that in alkali metals the rate of increase of mass is greater than rate of increase of volume, therefore the density increases down the group.