There are several ways to style the elements in each page, in Enterprise applications usually the CSS Framework size increased about 1 MB, and when your users a
File sizes
My first point would be that when dealing with an enterprise level application the actual total quantity of css when measured in megabytes is slightly less important, even for slow internet connections. It's important that the pages you load into an empty cache of a potential conversion that just clicked your pay per click ad for the first time are as tight as you can possibly make them, but for an app that a user is paying for and is intending to invest their time and effort, priming a cache every release, even with a megabyte of css is less of a problem. You could load it all last on the login page so it's all sorted while they put their credentials in.
Furthermore, you'll have the time to investigate some other techniques, such as loading critical 'above the fold' css in it's own, optimised file first; and splitting the css files up so that the common stuff is loaded on the first page view but any page specific stuff is loaded per page, as it's visited (for the record, this can be very good for the aforementioned PPC targets).
CCS Tricks goes into more detail here and here.
Complexity
One of the bigger considerations of enterprise cloud applications is the maintainability of the css. You're probably going to have a team of developers and a complex user interface. These things can quickly turn into a maintenance nightmare if the wrong decisions are made concerning the approach to css.
It's all very well if you users can load a page in 0.1s less, but if it takes you 30mins more to make every simple css edit then you're in trouble.
My recommendation
You want a combination of both. You should strive for semantic, context free css selectors in order to hit maximum re-usability (and low file size) and maximum maintainability. This allows for effective file size management and effective, scalable development.
For example:
.blue-box
.header-login-box
.contact-form-submit .green-button
bad: not semantic, or too context specific. I'm assuming that .blah
pretty much falls into this category, judging by the phrase 'do this for each element'.
.login-box
better: easier to re-use, semantic, but still too contextual
.box--highlighted
.button
.button--standout
even better: really re-usable because of complete decoupling from page context, but still clearly semantic, making it easier to maintain.
With the final examples you break your app UI designs down into modules which are defined and re-used wherever they are needed. It's conceivable that you may use more than one per HTML element, but you won't have ten.
It's also OK to use utility classes, such as .pull-left
in fact, Harry Roberts at CSS Wizardry, a successful consultant whose done this stuff in the wild for real clients recommends it.
Three further avenues of investigation
There are currently three organisational / naming strategies for scalable css architecture that try to tackle the problem, you might want to look at them in more detail:
BEM: docs introductory article
OOCSS: docs introductory article
SMACSS: docs and introduction
All three will help maximise re-usability and minimise file sizes while giving you rules to follow to keep things tight and help with new members of the team.