The phrase was discussed on Google Answers, here (a topic titled "Q: A Question only Bobbyd should answer" posted "23 Nov 2002 10:38 PST"), which identifies the phrase as coming from Epistles Book 1.
A copy of the Latin phrase is found here,
Isne tibi melius suadet, qui 'rem facias, rem,
si possis, recte, si non, quocumque modo rem,'
My literal translation (translating 'rem' as 'money'):
Is he to you better he who persuades, "make money, money, if possible, uprightly, if not, by any means money".
Another translation,
Is he better for you who tells you: ‘Make cash,
Honest cash if you can, if not, cash by any means,’
Did he write this?
Yes: Horace's epistles are well-known, and the literal translation of what he said is close to the idiomatic translation in the OP.
Did he intend this as advice, or is it quoted out of context?
It's quoted out of context.
Wikipedia summarizes the lines (by quoting summaries taken from a book) as,
41-69 - Men will do and suffer anything to avoid poverty, but they will do nothing to gain virtue, which is more precious than gold. A clear conscience makes a man truly a king.
Here is a translation of the whole verse,
Money or virtue?
Virtue is to flee vice, and wisdoms’ beginning is
Freedom from foolishness. See all your anxious thoughts
And risks to avoid what you deem the worst of evils,
Too meagre a fortune, some shameful lost election:
Eager for trade you dash off to farthest India,
Avoiding poverty with seas, shoals and flames:
Why not listen to, learn to trust, one wiser than yourself,
Cease to care for what you foolishly gaze at and crave?
What wrestler at village crossroads and country fairs
Would refuse the crown at mighty Olympia,
Given the hope, the prize of a dust-free victor’s palm?
Silver’s worth less than gold, gold’s worth less than virtue.
‘Citizens, O Citizens, first you must search for wealth,
Cash before virtue!’ So Janus’ arcade proclaims
From end to end, this saying old and young recite
Slate and satchel slung over their left shoulders.
You’ve a mind, character, eloquence, honour, but wait:
You’re a few thousand short of the needed four hundred:
You’ll be a pleb. Yet boys, playing, sing: ‘You’ll be king
If you act rightly.’ Let that be your wall of bronze,
To be free of guilt, with no wrongs to cause you pallor.
Tell me, please, what’s better, a Roscian privilege,
Or the children’s rhyme of a kingdom for doing right,
Sung once by real men like Curius and Camillus?
Is he better for you who tells you: ‘Make cash,
Honest cash if you can, if not, cash by any means,’
Just for a closer view of Pupius’ sad plays,
Or he who in person exhorts and equips you
To stand free and erect, defying fierce Fortune?
It's clear that he's not advising people to make money, whether correctly or otherwise: he is giving that as an example of bad advice.
Does 'make' here mean 'Increasing the amount of wealth in the world through productive work'?
No:
Horace thinks it's a bad/dishonourable activity, whereas Ayn Rand's meaning of it is as something to be proud of
The next set of lines include what should presumably be understood as the "cash by any means" method of making money:
What should I follow
Or whom? Some are eager for civil contracts: some
Hunt wealthy widows with fruits and titbits, or catch
Old men in nets to stock their reserves. With many
Interest quietly adds to their wealth.
It's clear that the "making money" described by Horace should not be understood as an honourable and productive activity described by Ayn Rand, i.e. it has almost the opposite 'meaning'.