"A manifold with boundary has dimension at least 1" if it has a dimension and if it has nonempty boundary?

I think Tu’s statement is fine:

A manifold, by definition, always has a dimension. Where are the charts going?

Usually when we say that a manifold “has boundary,” we mean that it has nonempty boundary.

After looking at some of Tu’s (non-standard!) definitions, I think you’re correct. An accurate statement might be

If an $n$-dimensional manifold has nonempty boundary, then $n\ge 1$.

Assuming sensible definitions, an alternative solution is to change the statement to the following:

A connected manifold with non-empty boundary has dimension at least 1

Edit: I rejected the suggested edit to change "manifold with non-empty boundary" to "manifold with boundary with non-empty boundary" because it does not add new information. A manifold with non-empty boundary must be a manifold with boundary, or your definitions are nonsense.

I would not say that a manifold could be dimensionless. A manifold consists of connected components, each of which has a dimension. As for the statement in question, a more accurate phrasing would be

"If an n-manifold has nonempty boundary, then $n \ge 1$"

or

"A connected manifold with nonempty boundary has dimension at least 1"

as was pointed out above by various commentators.