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The following is the Uniform Boundedness Principle and its short proof (using Baire Category Theorem) in Folland's Real Analysis:

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Here is my question:
Is there any particular reason that the "closed" ball $\overline{B(r,x_0)}$ is used in the proof instead of the open one $B(r,x_0)$?

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    You can do the same with the open ball. They've taken two steps: they have an open ball in one of the closed sets, and the closure of that ball must be contained in the closed set. They require less writing, but can cause confusion. – 2017-01-14

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One could indeed work with open sets instead. In Stein and Shakarchi's Functional Analysis, an almost the same argument (page 167) with open sets are used as the following.

By the assumption of (a), some $E_n$ must have non-empty interior, say when $n=M$. In other words, there exists $x_0\in X$ and $r>0$ so that $B(x_0,r)\subset E_M$. Hence for all $T\in\mathcal{A}$ we have $$ |T(x)|\leq M\quad\textrm{whenever }\ \ \|x-x_0\|