What is the secret to baking bread with a very fine crumb?
A couple of things will help give you smaller holes:


Keep the hydration reasonably low (say, 60% with American-style bread flour).
Use some oil or butter. Try 10% (baker's ratio).
Knead very well, something like 10–15 minutes in a stand mixer.
After the first rise, normally you try to be gentle, and not press out all the air. Don't. Instead, press it out.
Bake in a moderate (say, 350°F) oven. You don't particularly want much oven spring here.


For even finer texture, part way through the second rise, you can press it out again, and let it start a third rise (but not to double).

Also, adding some whole wheat or rye flour will give a denser crumb. With different flavor, too, of course.