You can take a look at the functionality matrix Functionality Matrix for ArcGIS 10 and Server | ArcGIS Blog to see if the specific task you are trying to do is supported by the Basic level or not. I know there are some restrictions with what you can do with "Distributed Geodatabases" and multi-user databases, but I agree with Darren, you should be able to add a field....but it does depend on what type of data you are working with. (rasters and images may have license level restrictions). Are you using Spatial Analyst with your Standard license for example....may be the extension that is needed and not the level.
Also, maybe there is a data lock and therefore schema lock on the data that could block it (can not change schema if anyone is in the file or if it is being used). things like that. Also, if it's versioned, I recommend rolling it back (i.e. reconciling and getting rid of versions) and getting the status ID = 0 before changing a schema. This might be more than you are dealing with, but wanted to mention.
btw - typically you shouldn't mark your own response as "helpful" unless it has something that is helping resolve the issues or will help others.