Indeed, there is no natural support for boundaries in Manopt or in the classical theory of Riemannian optimization.
One way is to use nonsmooth / constrained optimization techniques, as Ronny mentioned.
Sometimes, it's possible to smoothly parameterize your manifold with boundary, in which case you can optimize through that parameterization instead (and Manopt would apply).
For example, the simplex in R^n (which is not smooth) can be smoothly parameterized by entry-wise squaring the vectors on a unit sphere. That's a smooth parameterization that allows you to move any optimization problem on the simplex to an optimization problem on a sphere.
Likewise, you can smoothly parameterize the (nonsmooth) set of matrices of rank <= r using a factorization, as (L, R) -> LR*, where L and R have only r columns. (Notice that the set of matrices of rank = r is a smooth manifold, and can be handled directly in Manopt).