We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $α$ and $1-α$, respectively) and starting from the origin. We show that, almost surely, the asymptotic proportion of strings of $k$ consecutive visible lattice points visited by such an $α$-random walk is a certain constant $c_k(α)$, which is actually an (explicitly calculable) polynomial in $α$ of degree $2\lfloor(k-1)/2\rfloor $. For $k=1$, this gives that, almost surely, the asymptotic proportion of time the random walker is visible from the origin is $c_1(α)=6/π^2$, independently of $α$.