It makes no difference. Starting with a capital of 1, let $X_i$ be the multiplying factor for the $i$th day, so $X_i\in\{1+r,1-r\}$ with each possibility having probability 1/2. The expected capital after one day is
$$\mathbb E(X_1)=\frac12((1+r)+(1-r))=1.$$
After $n$ days, your capital is $X_1X_2\cdots X_n$, and
$$\mathbb E(X_1\cdots X_n)=\mathbb E(X_1)\cdots\mathbb E(X_n)=1$$
since the days are independent.
Clarifying notes
"$(1+r)(1-r)=1-r^2<1$ so as the stock goes up and down, I lose money"
but note that if the stock goes down, then down again, you have $$(1-r)^2=1-2r+r^2>1-2r,$$
so you get more than what you'd get using a "simple interest" idea, and this together with the up/up case cancels out the loss in the up/down case.
Indeed, for $n=2$ the expected capital is
$$\frac14[(1+r)^2+2(1+r)(1-r)+(1-r)^2)]=1.$$
Of course if you like or dislike risk then you may want to buy or not buy, respectively, but the expected capital remains 1.
On the other hand if the multipliers were not $\{1+r,1-r\}$ but $\{1+r, \frac1{1+r}\}$ then it would make sense to buy, as the expectation after one day would be $$\frac12\left(1+r+\frac1{1+r}\right)=\frac12\frac{(1+r)^2+1}{1+r}=1+\frac{r^2}{2(1+r)}>1.$$
