Does the unit step response not affect the transfer function?
this question says that this figure shows three unit step responses and was asked to find the transfer function. 


The picture you include is a single step input response, maybe there are different plots nearby with the other two step responses.


  Doesn't the input step responses only affect the output? 


Not sure what you mean by this, but for LTI system, you usually find the step response by having a system that is at zero \$y(0^-)\$ and with \$u(0^-)\$ a zero input before \$t=0\$, which then "steps" to be \$u(t)=1, \; t\geq 0\$. So yes, the output only depends on the input.


  So to look for the transfer function does not matter how many step unit response?


Again, you could be a bit more clear on your question, but if you know the input \$u(t)\$ and the output \$y(t)\$ you can find the transfer function \$H(s) = \frac{Y(s)}{U(s)}\$. 

So, suppose that by "having three step inputs" you mean that you had one step at t=0, another at t=1 and a last one at t=2, that would definitely not look like a single unit step input, nor like a single 3-units step. But, due to linearity and time invariance, \$y(t)\$ would look like having the unit step response \$g(t)\$ and some shifted versions of it, that is,  \$y(t)=g(t)+g(t-1)+g(t-2)\$.