The dimensionality of $| \psi \rangle$ doesn't have any physical significance (though you can formally set it to be whatever you want). The space of states is not the Hilbert space of the theory itself, but the projective Hilbert space. That is to say that two nonzero vectors in the Hilbert space $| \psi \rangle, | \psi' \rangle$ define the same physical state if $| \psi \rangle = c | \psi' \rangle$ for some nonzero (complex) scalar $c$.
When we talk about dimensionality in physics, it's talking about how quantities scale when we change the units that we use to describe the problem. That is to say, for example, if under a change from kilometers to meters (keeping all other units the same), some quantity $q$ changes numerically as $q \rightarrow 10^9 q$, then $q$ has dimensions of volume since it scales as length cubed.
Under the same change, how $| \psi \rangle$ changes is not a measurable quantity. It could change to $| \psi \rangle$ or $10^9 | \psi \rangle$ or even $ i| \psi \rangle$, but all of these are exactly the same physical state. So in quantum mechanics, the dimensionality of $| \psi \rangle$ is a formally meaningless quantity. It has no physical significance. There is no experiment that can measure it even in principle. You could choose a particular convention, but whatever choice you make has absolutely no impact on physics. 28
If you are going to choose a dimension, the most natural one is dimensionless.
With that said, it is often conventional to adopt some normalization condition for $| \psi \rangle$. This is done to facilitate explicit calculation, though it's not exactly physical. Such conditions can lead to some constraints on dimensionalities. One example is $\langle \psi | \psi \rangle = 1$. This requires $[\langle \psi |] = - [| \psi \rangle ] = -[D]$ where $[ \cdot ]$ denotes the total dimensionality (some might argue that this is true by definition of the dual space, but I won't go that far) and $[D]$ is just some dimensionality. If we require the 1-dimensional wavefunctions $\psi^(x) = \langle \psi | x \rangle$ and $\psi(x) = \langle x | \psi \rangle$ to satisfy $\int \psi^(x) \psi(x) dx =1$ then this gives us conditions for the position basis and dual basis, namely that $[\psi(x)] = [\psi^*(x)] = [L^{-1/2}]$, where $[L]$ denotes length. This gives $[|x \rangle] =-\frac12[L]+[D]$ and $[\langle x |] = -\frac12[L]-[D]$. We still have this $[D]$ floating around though, which is totally arbitrary, and the only way to get rid of it is to just pick some dimensionality for one of these quantities (or impose some equivalent criterion, such as requiring that $[\langle \psi |] = [| \psi \rangle ]$, giving $[D]=0$). No physical argument can give us a dimensionality $[D]$ for the state vector itself, as it has no physical meaning. 976