What did the founding fathers of America think about immigration?

I've deferred answering this because it is a complicated subject, and I can't find the right sources. My impression is that the founding fathers didn't share a coherent opinion on the subject; different states and their respective founding fathers had different opinions.

However yesterday, I heard the following paragraph read aloud

He has endeavoured to prevent the population of these States; for that purpose obstructing the Laws for Naturalization of Foreigners; refusing to pass others to encourage their migrations hither, and raising the conditions of new Appropriations of Lands. Declaration of Independence

The founding fathers publicly pledged the lives, their fortunes and their sacred honor to the cause of expanding immigration and naturalization of foreigners.

That said Pennsylvania was unique. I don't have good sources on this, but I believe Pauline Maier discusses it in her history of the ratification of the constitution, but Pennsylvania was, as far as I can recall, the only state that was in danger of not being English. (that's an oversimplification; rural PA was full of German speaking immigrants. Philadelphia, at the time the largest US city, was cosmopolitan but distinctively British. That said, the structure of the PA government was distorted by the fractious(update German) minority. If I recall correctly, one of the tactics used to force through a rapid ratification of the constitution was to not print it in German).

This touches upon a really fascinating cluster of debates in the history of the late colonial period and the early republic. There are likely many publications on this but it forms one of the central issues in:

• Aristide R. Zolberg A Nation by Design: Immigration Policy in the Fashioning of America

I'll focus on Zolberg's take. The book opens a discussion on this question with the same Declaration of Independence quotation (see @Mark-C-Wallace's answer) criticizing the English King for having "endeavoured to prevent the Population of these States" and thwart efforts to "encourage their migrations hither" but then emphasizes the importance of the conflict that this concealed:

Rather than isolated skirmishes, the confrontations over these issues were vital episodes in the larger war over sovereignty, and amounted to an epochal struggle over the structure or “design” of American society. (p25)

Zolberg argues that the references to immigration in the declaration probably is referring to the disallowance of a North Carolina act of 1771 and an Order in Council of 1773 preventing migration into recently-French territory (p25). However, Zolberg points out, they were also angry at the crown for the opposite: British efforts to prevent the colonials from barring "undesirables" such as convicts and paupers, or limitations on the slave trade (p26). On the other hand, there was equal enthusiasm for excluding Catholics for the most part (p37). Zolberg summarizing the main differences among colonials and the British thus:

Both sides had similar notions regarding what sorts of people were desirable and undesirable; but whereas Britain was intent on ridding itself of convicts and paupers while seeking to retain the conforming and productive, the colonists were equally adamant to keep out the first and attract the second. Both sides shared a mercantilist understanding of population as the major source of wealth and power; but whereas this led Britain to try to keep the colonial population within bounds, it prompted the Americans to maximize their num- bers by all possible means. (p40)

As you point out, Franklin was no fan of German immigration, but also this issue of undesirables or British "emptying their Jails into our Settlements" (quoted in p41). Zolberg points out that he, and later Jefferson would both embrace anti-immigrationist arguments because they believed that "natural Generation" will eventually make up any temporary deficit in population in a place like the colonies (p45). But on the other hand, he opposed British restrictions, arguing that "law is unnecessary because population tends toward equilibrium" (quoted in p46) and that everyone has a natural right to leave their place of abode (again a view shared by Jefferson).

In contrast, Alexander Hamilton argued in his “Report on Manufactures” of 1791 that support for industry would have the desirable effect of drawing more immigrants, some of whom would abandon this to fill the needs for agricultural labor (p69). In the meantime, the use of immigrants in factories would "Americans free to carry on more dignified activities" thus,

The leader of the Federalist Party, which within a few years of the “Report” would enact the notorious Alien and Sedition Acts that established it as the fountainhead of “nativism,” thus also deserves credit as the first explicit advocate of the mass immigration... (p70)

On other founders and related issues:

• George Washington proposed importation of German labor to District of Columbia in 1792 (p72)
• After condemning the British for dumping convincts on the colonies, Benjamin Franklin proposed exporting American convicts to Scotland (p73)
• Tom Paine was confident that, despite its diversity, a justly constructed government could ensure union (p87)
• Tench Coxe emphasized the urgent need for assimilation of new immigrants (p83) and warned that foreign powers could use emigration as a weapon (p84)
• The 1788 election German pattern of voting as a bloc created a "feedback loop" which pushed open views on immigration (p85)

Many of these points and others also contained in a shorter 1994 article in Policy Review by Matthew Spalding found here:

• http://www.thefreelibrary.com/From+pluribus+to+unum%3A+immigration+and+the+Founding+Fathers.-a015028217

Overall, it seems like, both in Zolberg and in other works, the early Republic and late colonial debates on immigration reach a turning point with the important 1798 Alien and Sedition Acts, as the discussion turns to the "rise of nativism" in the United States.

Some other sources:

• American Immigration: A Very Short History, p17-25
• Edward P. Hutchinson Legislative History of American Immigration Policy 1798–1965
• Marilyn C. Baseler, “Asylum for Mankind”: America, 1607-1800
• A.G.Roeber,“‘The Origin of Whatever Is Not English among Us’:TheDutch- Speaking and the German-Speaking Peoples of Colonial British America,” in Strangers within the Realm: Cultural Margins of the First British Empire
• Frank George Franklin, The Legislative History of Naturalization in the United States: From the Revolutionary War to 1861
• Matthew Spalding “From Pluribus to Unum: Immigration and the Founding Fathers,” Policy Review 67 1994