I am trying to get cells in column
C to turn green when the cell in the same row of column
B has a value in it.
You can use conditional formatting with a custom formula like
=NOT(ISBLANK(B2)) across the range you want to use, in this case C2:C60. Make sure not to use any $ signs and to reference the first cell in your range (in your screenshot there's an offset), it should be B2.
Another reason why your set-up may not work is because the top-most conditional formatting rule that is true rules and overwrites other rules. So it could be that your red background is in front of the green.
To investigate particular circumstances you would have to make available. Here is mine: https://docs.google.com/spreadsheets/d/1KKjaSzsTc4gd1gwobg775jZoVzDK9BeGhv3xOzKJiiY/edit?usp=sharing
Here's a picture of the set-up matching yours apart from the red background: