The function is definitely computable. It is either the constant $1$ function, if there are arbitrarily long sequences of consecutive $1$s in the expansion of $e$, or if there is not, then the function is a simple step function, with value $1$ up to some value $k$ (the largest sequence occurring), and $0$ thereafter. All such functions are computable, and therefore, your function is computable. 

One shouldn't confuse the fact of the computability of your function with our lack of knowledge about which computable function it is.

See also [my answer to a related question](http://mathoverflow.net/a/48031/1946), where a similar point is made.