Well, I use firefox, because it is
just associated with Mozilla. However, with recent updates, it started to widen it's data-harvesting behaviour: The new "New tab" page by default includes "suggested" and "sponsored" sites. In clear text, your browsing history is analyzed and matched against a database. This behaviour can be disabled, and combined with a few tweaks in firefox' privacy settings it is probably a better choice. However, the question remains:
For how long? In that sense, using text-mode browsers might be a viable option privacy-wise, but probably doesn't qualify as modern to many. There are lightweight browsers such as Midori, Dillo, NetSurf... that are GUI-based, as well as some keyboard-centric ones. Those tend to use rendering engines the major browsers use as well, mostly Gecko and webkit, so page rendering should be an ok experience. However, community (if that matters), plugins/addons, and similar stuff aren't a thing there.
TL;DR: It's a compromise between modernism and privacy, and that sucks. We should use the smaller, free browsers more to support their communities and improve them to get out of this dillemma. Hoever, this requires will and work.
EDIT: Now, your question got me interested, so I decided to do some research and do two things:
1. Gather links of minimalistic browsers (primary keyboard-centric)
2. Write some short reviews / report back what I decided to use.
Alas, the links:
*
Opera - I used it as a child, pretty mainstream, but a nice thing.
*
dwb - webkit based, minimalistic.
Notes: I installed it on Arch from the repos, and got a segfault pretty fast.
*
Vimprobable - seems pretty vim-ish, but kinda heavy maybe
*
jumanji - From the makers of the zathura document reader (which I can only recommend)
Notes: not in production yet.
*
luakit - Configured in lua, webkit-based
Notes: seems awesome and very powerful, trying it out now.
*
Conkeror - pretty well known as well
*
uzbl - browsing
tools adhering to the UNIX philosophy
I hope this cleared the situation up a bit.
