The exponential function $f(x)=e^{-x}+1$ crosses the $y$-axis at which of the following points? $(0,1)$ $(1,0)$ $(2,0)$ $(0,2)$

Step 1 ：The function $f(x)=e^{-x}+1$ crosses the y-axis when $x=0$. We need to substitute $x=0$ into the function to find the corresponding y-coordinate.

Step 2 ：Substitute $x=0$ into the function: $f(0)=e^{-0}+1=1+1=2$

Step 3 ：So, the function $f(x)=e^{-x}+1$ crosses the y-axis at the point $(0,2)$

Step 4 ：Final Answer: \(\boxed{(0,2)}\)