In general from what I've seen problems with NoSQL databases are somewhat different to standard SQL, generally as a result of their focus.
First up is that in general many NoSQL databases lack a lot of security features (e.g. fine grained authentication/authorisation). The websites of the products themselves will actually say that they are only designed to be accessed from "trusted environments" (e.g. redis, and mongodb).
How important their limitations are, would likely depend on what you want to use them for. So a standard web application may be ok (only one level of access needed and you can retrict access at an IP level to just the webserver), but a system designed for multiple users with different privileges to directly access the database would be problematic.
Secondly (and I accept this point could be seen as a bit FUD'y) there's maturity of the codebase. These products are all relatively new and have grown up quite quickly, so I'd expect that there has been less scrutiny on them from a security standpoint compared to engines like Oracle DBMS or MS SQL. Whilst both those DBs have had a lot of problems in the past, more recently (especially in the case of MS-SQL) they've had a pretty good security record.
The flip-side of this is that the codebases are a lot smaller and the products much simpler so there's less functionality to attack. Also as (AFAIK) they lack features like the ability to execute operating systems commands, there's potentially a lower impact of compromise than you might see with the big relational DBMS' which have huge ranges of functions available.