One thing you need to consider is how much humidity is in what you have before you put it the plastic tubs. That can vary depending on the time of year, and can be deceiving. Where you store the sealed tubs matter as well. If they are somewhere where they get hot in the summer, you can have mold issues because they don't breath...especially with items that were not stored well in the past and might have spores just waiting for the right moment to activate.
If you were going the full museum route, you would want acid free boxes stored in a temperature and humidity controlled environment.
However, that is not practical for most of us. I think most people use a combination of things depending on what they are storing and where it will be stored. I have both some rubbermaid tubs that seal and some flip top plastic storage boxes, plus an assortment of cardboard boxes. One issue with the bigger rubbermaid tubs, is over time, if you put too much weight in and stack them, the sides of the bottom tubs can collapse.
In my experience, if you are storing things inside your house, some air circulation is a good thing, unless your items are in a damp place. Lastly, just about anything you store needs to be looked at occasionally. If you use plastic tubs that seal, you should plan to air the items out, picking days that are clear and dry. That might just be opening the lid for a while, or it might be taking everything out for an airing.
There is not an exact answer to your question. I think storage is an ongoing issue that is always evolving, but you will eventually hit on a combination that best suits your conditions and needs.