If the base is bulky E2 will be favored. Bulky substrates by definition favor E2 over subsitution also because
(1) They are not open for nucleophillic attack and
(2) they form more stable alkenes.
Don't think in terms of bulky though, remember it's in reference to how substituted the carbon is. So, Sn2 rxns favor primary carbons (ie, methyl carbons), and E2, Sn1 and E1 favor tertiary carbons. While the ones that need to be considered when trying to decide between Sn2 or E2, E1, Sn1, are in relation to the secondary carbons.
Also, Reaction temperature can be a good indicator as well: If you have a temp <50 degrees it is likely either going to be Sn2 or Sn1.
If the temp is at Reflux or very high, it is likely going to be E1 or E2.
Also, I believe your confusion about the base comes about because it doesn't seem like you're considering the solvent.
So, for Sn2:
Nucleophile: Strong
Base: Weak
Solvent:
Aprotic (ie, DMSO or DMF; Think RO
R)
For E2:
Nucleophile: Weak
Base: Strong
Solvent: Protic (Think RO
H)
Also, remember that basicity is dependent upon the type of solvent. So, what is a weak base in one solvent can be a strong base in another.