Having read Frankenstein recently it's pretty clear Shelley had a weary of the powers and ethics behind scientific advancement brought on around the Industrial Revolution, but was this view popular? Did many people object to it?
Very interesting question -- and difficult to answer since it was a period of considerable change in science itself; the word "scientist" itself only dates from the 1830s, for example. Since your question brings up Shelley, I'll focus on the British context and try to explain the background of thinking that led to the perspective on industrialisation you might get out of Frankenstein. Shelley brings up particular problems, though, because of the way in which Frankenstein has been used and abused by modern interpreters, who've often read it in reductive ways. It's not simply a work about the dangers of science. Like Steven E. Jones points out,
reductive readings of the novel as crudely “antiscience” or “antitechnology” overlook details that make significant differences. Victor Frankenstein’s failure is a failure of imagination, sympathy, and responsibility for his creation, but it is not at all clear that the novel simply condemns him (or Walton, the explorer whose expedition so closely mirrors Frankenstein’s own pursuits) for seeking or applying “forbidden” knowledge, let alone “technology” in the modern sense. (Jones, p. 110)
As always with these kinds of broad questions I'll also give the proviso that my interest is in the history of political thought, so I will put some focus on the specifically political side of things, though as I'll explain I think it's unavoidable here.
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On that note, let me start by pointing out more generally that if we want to consider the reception of industrialisation and the perils of industrial scientific advancement, we're really looking at a debate not so much over the principles and philosophy of science in itself, though that certainly figures into it, as over the effects of technology and the principles of political economy. The big questions in this controversy were to do with commerce, the role of technology, manufacturing, industrial machinery, and not so much with inquiry into the natural world as such. There's more than a grain of truth when people nowadays point out that a "mad scientist" like Victor Frankenstein is really more of a mad engineer!
So let's try and grapple with the state of thinking on these issues at the turn of the 19th century. Scepticism about industrialisation drew on strands of thinking that had been prominent in the previous century. The French Physiocrats, who exercised a great (if often reactive) influence on British political economy and the thinking of people like Adam Smith, had argued at this time that the only true source of wealth was the land: thus in the Encyclopédie in 1757, François Quesnay had claimed that "commerce is merely a branch of agriculture" (Encyclopédie, s.v. "Grains"). When put alongside the thinking of someone like Jean-Jacques Rousseau, who was famously sceptical of the arts and sciences and expounded at length on the vices of civilised society (see his First and Second Discourses), it's easy to see how this could later make for a powerful critique of industrialisation.
While the Physiocrats and Rousseau themselves fell out of fashion -- the context became especially problematic in Britain given the fight against the Revolution and Napoleon -- the same kind of thinking still persisted at the turn of the 19th century in Britain. In his 1808 work Agriculture the Source of Wealth in Britain the economist William Spence, though he was careful to detach himself from the Physiocrats, argued strongly for the role of agriculture against what he saw as the frivolities of manufacturing and commerce. Spence's work was controversial, and debated at length by various of his better-remembered opponents, like James Mill and Thoms Malthus.
In the political sphere, it gained a good deal of purchase thanks to the deleterious effects of early industrialism on British workers. Drawing on Spence, figures like the radical pamphleteer and later MP William Cobbett denounced the proliferation of manufacturing as a means of swelling the pockets of the rich at the expense of the poor. Combining the economic critique of manufacturing with an older current of British radical scepticism of commercial interests, Cobbett adopted a slogan from the Whig William Windham: "Perish commerce, and let the constitution live!"
In the 1810s, this economically grounded opposition to manufacturing combined with the growth of popular discontent in parts of Britain to produce what we now remember as Luddism. The Luddites were (as Jones points out, see my sources below) not just opponents of technology, and weren't simply concerned about machines automating away their jobs -- though that was certainly a part of it. They were a radical political movement, whose famous acts of machine-breaking were "controlled and selective" rather than willy-nilly (Archer, p. 56), designed to apply political pressure rather than simply being vandalism against technology.
This was the politically-charged context in which Shelley wrote Frankenstein in 1818. It's important to recognise here, again, that scepticism of industrialisation did not necessarily mean scepticism of technological advancement as such. Percy Shelley, the author's husband, was certainly by our standards a man of science himself, with experience in scientific experiment in his youth, but he was nonetheless sympathetic to Luddism. In 1812, when the military suppressed the Luddites in Nottingham, Percy Shelley wrote a letter cursing the military "if they destroy one of its famine wasted inhabitants", and he was generally well-known for some of his radical views.
The Shelleys and many of their circle were also opposed to specific Luddite methods, however. This is vital to understand what Mary Shelley thought she was doing with Frankenstein. As Adriana Craciun points out, Frankenstein had a direct and obvious bearing on this immediate political context: keenly aware of the struggles over Luddism and the continuing aftereffects of the French Revolution, Shelley was "an informed, critical observer and liberal sympathizer who wishe[d] to prevent both continued injustice and revolutionary violence", and the narrative of Frankenstein reflects this -- Frankenstein's monster has been interpreted as an allegory for revolutionary violence, among others. (I think the use of "liberal" in Craciun's quote unhelpful, but that's a story for another time -- the general point still stands.) Shelley's views weren't popular to the extent that they were at a distance from both the radicalism and the conservatism of the period, but they were widely applicable.
That's the political context. Now it's often supposed that a movement called Romanticism arose in the early 19th century as a literary and artistic critique of science. But scepticism of industrialisation among the canonical Romantics was never quite as clear-cut as it's sometimes been made out -- it wasn't a question of scientific luminaries facing off their Romantic opponents. William Wordsworth, for instance, is often remembered as an opponent of the social changes brought about by industrialisation, but he was just as impressed by its technical advances:
I rejoice, / Measuring the force of those gigantic powers / That, by the thinking mind, have been compelled / To serve the will of feeble-bodied Man (The Excursion 8.204-207, published 1814)
For that reason, and for the more general reasons I set out at the beginning, I'd encourage you to leave the idea of "science and its critics" aside -- for this period at least. This narrative is bound up with all sorts of other pieces of problematic historiography, like the triumphalist popular historiography of science, or ideas like the Berlinian "Counter-Enlightenment", which in many scholars' opinion tend to be more misleading than helpful. The currents that influenced Frankenstein, and the scepticism of industrialisation that it reflected, came out of a specific political-economic critique of technology that reached a climax in the early 19th century.
When you ask whether many people objected to it, then, the simple answer is that there was little to object to -- the Romantics and Mary Shelley were not waging any kind of war against science. They were measured supporters of industrial advancement, who sympathised with movements like the Luddite resistance and the intellectual critique of commerce that surrounded it, but disagreed with their politics.
Sources