How can I- be both a nucleophile and a weak base?
As Jori pointed out, nucleophilicity is a kinetic property while basicity is a thermodynamic property. Nucleophilicity in part is about the rates of chemical reactions; thermodynamics is in part about the favorability of chemical reactions. 

A reaction may be very favorable, i.e. it has a very negative free energy change, but it might take a long time to actually complete. For example, the rusting of iron is favorable from a thermodynamic standpoint. The products, iron oxides, are very stable. However, the process is relatively slow from a kinetic standpoint; putting water on iron isn't going to instantly result in a familiar reddish-brown hue. It might take a day or so. 

On the other hand, if you squirt something such as tert-butyllithium into the air, you'll probably catch yourself or something on fire. This is a carboanion salt. Carboanions are strong (Bronsted, and therefore, Lewis) bases. Acid-base reactions also tend to be relatively fast (kinetics) since they involve the transfer of something so small - a hydrogen proton. A graduate student unfortunately exposed some of this carbocation salt to the air and the resulting fireball instantly caught her sweater on fire. The carboanion salt likely grabbed protons from atmospheric moisture - a very favorable process from a thermodynamic standpoint - and kinetically, a very fast process. And so the resulting heat likely ignited the resulting protonated carboanion (alkane in this case) on fire. 

In your case, $\ce{I-}$ is a good nucleophile in both protic and non-protic solvents. It's a rather big ion - the biggest of the halide anions. That means its negative charge is relatively dilute. This in turn means it's not as strongly solvated by the solvent molecules; it doesn't have as many tenacious solvent molecules hindering its trek toward a carbocation. So $\ce{I-}$ can reach carbocations relatively quickly. Note that terms dealing with time usually indicate that the writer is speaking about a chemical's kinetic properties. 

For the same reason that $\ce{I-}$ is a good nucleophile it's a poor base. It's dilute negative charge (due to its size) means that $\ce{H+}$ has better things to do than be attracted to $\ce{I-}$. $\ce{I-}$ has a dilute amount of negative charge so it cannot effectively stabilize the hydrogen proton as well as many solvents might - especially water, which extensively associates with $\ce{H+}$; $\ce{H3O+}$ only begins to describe how well the proton is associated with water molecules; the proton has been found to be associated with 2, 3, 4, or more water molecules!