As a small business owner, you would have recourse for dealing with deadbeat customers. A guy like Pip or Rawdon Crawley wouldn't be allowed to fleece you forever, unless you let them--and if Sir Richard Dastardly was rich enough, sometimes you would just let it slide for a little bit. If they "paid you in exposure" and sent their (possibly more solvent) friends to you to buy suits or dresses, you might let it slide for a little while, just to be neighborly and keep the peace. But if you were doing House of Worth work for pawnshop pay, you'd eventually get fed up. You'd start sending duns/debt collectors to knock on their door, hoping the annoyance and potential shame would convince them to pony up. At this point, he might contract with a moneylender for a promissory note; the lender would pay you and charge Dastardly interest, and Dastardly would be out of your hair.

If your debtor doesn't find a way to pay you, your next step is prosecution. You want Sir Richard to actually show up for the trial, so you bring legal action against him and pay a shilling for an arrest warrant. A sheriff's officer serves the warrant and will take Sir Richard Dastardly to a sponging house. There, you have some time to settle or you'll end up on trial/in jail.

If it gets to trial, and Dastardly is found to be a debtor, they stick him in jail. You can't just take his stuff, however, not yet -- he still has to decide to pay you, by selling his own stuff or borrowing money from family and friends. Rawdon Crawley never gets to this point, but a number of Dickens characters did. Usually, if you ended up in debtor's prison, you were already poor--not a Sir Richard Dastardly. Typically, a person with a title would have resources: they could sell land, a horse, beg Auntie for cash, or marry some wealthy heiress. If you were poor and in debtor's prison, you did have some light at the end of the tunnel: you could still ply certain trades from within prison, and you could eventually hope to get out. But it was miserable and difficult to drag oneself out of that debt hole. (Imprisonment for debt was abolished in 1869.)

Now, as to your question, "Did they care?" I think they cared about as much as we do. Victorians were a little more moralistic than we are now in that they were generally more likely to judge their neighbors as "bad" than put the circumstance ahead of their judgment -- but a lot of their attitudes are similar to today's. Being a serial debtor was more personal to the Victorians than it is to us, because there were, with the possible exception of the East India Company, no giant faceless monolithic companies to which you would owe money. You'd owe money to Bob the tailor and Joe the greengrocer, guys who lived probably within a few miles' radius of your own house, not to MasterCard. No one goes hungry if you don't pay MasterCard, but a Victorian person refusing to make good on his debts was literally taking money away from his neighbors, which was a bad look. Tongues would start wagging when the debt collectors came knocking, and a Sir Richard type might find himself not invited to as many parties as before. If he had romantic prospects, they could dry up: the father of a young lady isn't going to want his daughter hooking up with a spendthrift who'll end up in prison and will be unable or unwilling to provide for his family. If he's part of a gentleman's club, depending on how bad his money woes are (and whether or not all the other men in his club have similar attitudes towards money), he could find himself kicked out.

There were jerks, con men, liars, and people who had "f-you" amounts of money where they could gleefully skip town on Bob the Tailor and not care. There were no credit scores in the time period, but all you had was your name. If you developed a reputation as a sketchball, tailors wouldn't want to do work for you on credit anymore--no matter who you were. Tailors for the most part couldn't afford to give their work away for free. You'd have to pay cash. If you had cash, this wouldn't actually be a big deal, which is why a lot of wealthier people might only care about their debts insofar as their reputation could be affected. If Bob didn't want to play ball anymore, they'd just pay Bob the cash and get credit from some other poor slob. If you didn't have cash, this would be a serious issue for you. Some "wealthy" people didn't actually have "ready money," only large fixed assets like land or houses, which they would then have to mortgage or sell if they'd tanked their reputation/credit, or who were known to be out of money even if they'd never had a debt problem. Like today, a lot of "rich people" were cash-poor, and kept an appearance of wealth by leveraging what they did have to get more.

(Sources I looked at to write this: "What Jane Austen Ate and Charles Dickens Knew," by Daniel Pool; "The Victorian City" by Judith Flanders; "Vanity Fair" by William Makepeace Thackeray.)

EDIT: I corrected the naming. While “Sir Dastardly” sounded like a very interesting person to me, and is more descriptive of his character, the commenter below is right—it’s Sir Richard (Sir Dick if ya nasty).

In 1896 Lady Randolph Churchill received a letter from her son in which he described the state of his finances as “damned poor”. As a cornet in the smart (if flashy) 4th Hussars then stationed on tour in Bangalore, Churchill quickly found himself burdened by the standard of living expected by the regiment. As a junior officer, his annual salary amounted to approximately £120 per annum. This was not enough to pay his mess bills, that is, the cost of the lavish entertaining that all officers were expected to contribute towards on an equal footing, nor did it cover any of the other exorbitant costs that officers were expected to pay while on service. The initial cost that Churchill incurred upon joining the army was enormous:

Yet an incoming officer had a huge burden of both initial and on-going expenses. He had to pay for his own uniforms, civilian clothing, cases, swords, saddlery, spurs and other equipment, clothes for his personal servant, the furniture and decoration for his quarters, and various subscriptions for the regiment the officers’ mess and special events. To these were added mess bills for food and drink, the cost of two… chargers and whatever polo ponies he required. Even after his aunt [The Duchess of Marlborough]… bought one charger for him at £200, Churchill calculated the initial cost of equipping himself at £653.

While Churchill can somewhat be blamed for selecting a regiment with a particularly aristocratic mess, in truth many contemporaries of his class were increasingly concerned at the large amount of money that young officers were expected to spend beyond their annual income regardless of their regiment. Indeed, in the same year in which Churchill wrote his mother, he also had to witness two of his fellow junior officers bullied out of the regiment for their inability to pay. They had annual private incomes of £300 and £500 each. The matter was so severe, so ludicrous, that the incident was brought forth and debated in the House of Commons resulting in the Akers-Douglas Committee, which declared after much opposition that the private contributions of officers had reached a level where it was harming the officer corps: they found infantry officers could expect to spend anywhere between £150 and £250 per annum and cavalry officers somewhere in between £400 and £700. Even Queen Victoria was surprised to learn that a junior officer in one of the elite guards regiments required an income of at least £1,000 to participate in regimental life, more often somewhere in between £1,200 and £1,500 on average.

In comparison, a 1901 census found that the average worker in Britain made £43 a year. A lower middle-class professional such as a clerk was expected to make somewhere in between £150 to £200. A rich man would make £1,000. The extraordinarily rich, however, were deemed to have an annual income of over £10,000 per annum with a large house in London and a second in the country.

But unlike our modern system of consistent returns, this would not always be a constant revenue stream.

It was generally expected that merchants would extend credit for their goods and services and that, occasionally, this business would have to be structured on layaway. Certain merchants like grocers (cheesemongers, butchers, etc.) and wine merchants generally expected payment at the end of the calendar month. However, tailors and the like often had to wait much longer, in part due to the expense of the item. These larger bills would be paid, assuming some level of fiscal responsibility, at the end of the year or whenever a man with unsteady income knew he would be in possession of a large sum of money. In the interim, or when there was no option but to pay immediately, there were loans, often charged at exorbitant rates in order to justify the risk of the proposition. In the event that the bill was pressing, loans could even be made on an individual’s inheritance as collateral.

Bankers were therefore a frequent partner in the lives of many well-to-do people, although they were often (rightfully) viewed with some skepticism by their patrons. Lady Randolph Churchill’s healthy income of £5,000 a year was initially viewed as more than sufficient for her small family -- Winston and his brother Jack were her only dependents -- but her acquisition and renovation of a large Paris apartment, nominally to save money as Paris was judged cheaper to live than London, had only gotten her into inadvertent debt. Her financial counsel was likewise compromised, in part because her estate was so legally complex and bound by (then) sophisticated financial instruments. Reconsolidation was therefore an almost ruinous task that saw Lady Randolph flee back to England in order to live in a well-appointed country house north of London. She reported her early expenses as thus in 1896:

Out of £2,700 a year, £800 goes to you two boys, £410 for house rent and stables, which leave me about £1,500 for everything -- taxes, servants, stables, food, dress, travelling & now I have to pay interest in money borrowed

The Churchills would struggle between paying their bills and incurring new ones for the rest of their lives.

But what to do about a rakish gentleman? This question plagued tradesmen to no end. In some cases, the pretention of merchants who classified themselves as members of the “carriage trade” protected them from classes who could be classified as risky, but certainly there was always the debtor’s prison and lawsuits. Peers were far more notorious for skipping their bills than any other class, but the bankruptcy of the Duke of Newcastle in 1869 finally brought public outrage regarding the impossibility of imprisoning peers (but not baronets) while members of the body public could be arrested and imprisoned for financial mismanagement. The case was important because the duke, who had been living in France to escape his creditors, was under immense pressure to return to Britain after accruing an astounding £230,000 in debts. At the end of the day, a merchant simply had to accept that some bills for small(er) goods and services would simply go unpaid: it was good business to be perceived as the preferred choice of aristocrats and capitalists, even better if such cases brought some notoriety in exchange for little loss.

I answered a similar question (from the debtor's point of view) a while back, which you might find useful: https://www.reddit.com/r/AskHistorians/comments/7hau5e/in_poldark_18th_century_lenders_are_able_to_call/dqrq0iw?utm_source=share&utm_medium=web2x