how do I fix loose tile on basement stairs
If you are in the US and have a traditional stringer staircase leading to your basement there is almost nothing that you can do to prevent issues with a tile installation.  


There is too much movement in this type of staircase.
Dimensional lumber moves too much.
Unless there is a closed door at the top you have two climate zones hitting each other on the stairs.
Basement stairs are about as steep as possible in most cases causing more stress on the stairs.  


To get tiles to work you would have to add in supports under the stairs and into the framing on each side, put concrete board over stairs, possibly plywood under concrete board, and then a membrane.  And even then you might still have issues.  

Tiles look great on stairs if done right.  I have mentioned before that I have rehabbed houses in Europe.  I have gutted some 80 year old homes in France and never had to touch the tile floor/stairs - which after cleaning hold up better than a wood floor.  However the tile stairs were laid on top of concrete 100% of the time.   

Meaning unless you want to pour concrete stairs, don't tile.  You can put some nice bullnose pine treads on the stairs with some oil based paint that will blend in just fine to tile and will hold up and look nice.  Whatever you do to temporarily fix the issue is just going to cost you time/money/frustration.  The tiles are a sunk cost, sink them.