I can neither speak for nor against the involvement of James Roosevelt in the opium trade, although needless to say there are quite a few people whose fortunes were made in that area. (Regarding British traders, apart from the obvious Jardine and Matheson families, the Sassoon family – yes, that includes the WWI poet Siegfried – were a major player as well.) Americans who made it rich either directly or collaterally through opium included John R. Forbes, private translator to the Hakka tea merchant Howqua and who later became a major railroad magnate (although Howqua and John Forbes avoided direct involvement in the trade, Forbes' two brothers at Canton did not, and Robert B. Forbes in particular was one of the main American opium traders.)
But back to the matter of American involvement in the opium trade, whilst it is true that the U.S. was the second largest mover of opium to China, there are a couple of caveats to make. For one, American merchants were not carrying opium from American territory. They were either 'country traders' taking 'Patna' (a.k.a. 'Bengal' or 'Benares') and 'Malwa' opium from India to China, or, more commonly, picking up cargoes of 'Levant' opium from Smyrna (modern-day İzmir, Turkey). For another, this made the American trade a relatively distant second place to the British. Barely 8% of the opium arriving in China by sea was Turkish opium moved by American merchants. However, Turkish opium had a higher proportion of morphine (10-15%) than Indian varieties (typically 3-4%), so was somewhat distinct. Dikötter et al. suggests that the disconnect between the strange trips experienced by European consumers and the apparent lack of significant medical impacts on Chinese smokers may be down to both the reduced morphine content of the original product and the slower and more wasteful method of consumption – whatever the case may be, it the core of the opium trade was still British. Certainly, though, there was an increasing amount of Indian opium being moved by US ships in the pre-Opium War period, and it's certainly far from inconceivable, taking the case of John R. Forbes, that the elder Roosevelt may have made an investment in Chinese trade, though again as I am not familiar with his case I can say nothing more.
Your history professor is a bit off base on the familial aspect here: It wasn't FDR's father who was an opium trader, but it was his maternal grandfather, Warren Delano, Jr. A thorough account of Delano's short-lived career as an opium trader is detailed in the book The Private Side of the Canton Trade, 1700–1840: Beyond the Companies ed. by Paul A. Van Dyke and Susan E. Schopp. You can find another, briefer, account in The India-China Opium Trade in the Nineteenth Century by Hunt Janin. Warren Delano Jr. worked for the same company as the Forbes did that /u/EnclavedMicrostate mentioned, and through those connections, Delano was also a trading partner of the Chinese merchant Howqua mentioned in that reply.
But go back a bit further: the Delano family immigrated to Massachusetts on one of the first ships over to New England after the Mayflower. By 1776, the family had become one of the wealthiest in the area of Fairhaven, Massachusetts, having acquired considerable farmland and then splitting it up and selling it off. By the time of Ephraim Delano--FDR's great-great-grandfather and a Revolutionary War captain--the family was already rich.
FDR's great-grandfather, Ephraim's son, Warren Delano, Sr., was also a military man, and fought in the War of 1812, where he was taken POW. The family suffered financially when their property was damaged during the attack on the nearby Fort Phoenix by the HMS Nimrod on June 13, 1814, followed by a naval blockade that affected local commerce. The family suffered further losses when a "great gale" (probably a tropical storm, or perhaps a hurricane) hit Fairhaven on September 23, 1815.
Nonetheless, the family quickly recovered and, in 1832, Warren built a large 10-room home in Fairhaven, one of the largest in the town, and the home would remain in the Delano family until 1942. It is now a bed and breakfast. By this time, Warren Sr. had served several terms as a selectman for the town of Fairhaven, which was followed by a term in the Massachusetts state House as representative from Fairhaven, to which he was elected the same year he built the house. He was also instrumental in funding a new church in Fairhaven, which became the Unitarian Church.
His son, then, came from money. In 1833, at the age of 24, Warren Delano, Jr., went to China working as a trader for Russell & Company. As detailed in the aforementioned The Private Side of the Canton Trade, 1700–1840: Beyond the Companies, Warren Jr.'s working career with Russell & Co. in China began by doing legitimate business in the porcelain and tea trade. Warren made a bunch of money for the company, but he was not above some illegal opium smuggling, beginning a couple of years after he had arrived:
"In 1835 Delano's relationship with Houqua [a Chinese merchant who Delano enjoyed a profitable trading relationship with] was sorely tested as Russell and Co. began to supplement its legitimate trade in tea and porcelain with opium smuggling. Although British and other India-based merchants dominated the drug trade, Russell and Co. found a niche market in lower grade but cheaper Turkish opium...In March 1839, the Chinese emperor dispatched Imperial Commissioner Lin Zexu to Canton to put a stop to the opium trade, through force if necessary. Lin arrested 1,700 Chinese opium dealers and confiscated 70,000 opium pipes in just six weeks. He also demanded that Western merchants sign an oath not to participate in the opium trade and to register a bond with the Chinese government to ensure their good behaviour."
As the book goes on to say, Russell & Co. ships were a continued target of Chinese raids to stop the opium trade, even though Delano had put a stop to their opium smuggling for fear of getting caught. A blockade ensued, which effectively stopped the Russell & Co. ship the Cambridge from leaving the port, so Delano sold it to the Chinese Navy, but he refused to put up the money for the bond demanded by the Chinese government. The First Opium War broke out shortly after that, and Delano engaged in war profiteering during the fighting, though probably not any further drug smuggling. He left China in 1843, shortly after the First Opium War ended. In all, through the various trading that Delano had done during his decade in China, which included about four years of opium smuggling among more legitimate activities, he had made a fortune of $200,000.
Still, this was just a fortune of the Delano family, and the family had already been rich before this had all happened. After Delano returned to the U.S., he continued to work for and make money for Russell & Co. before retiring and living off of other investments, including a profitable stock portfolio.
Further, this was only one lineage that Franklin D. Roosevelt came from. On his paternal side, the family had been extremely wealthy since at least the 1600s, beginning with the very first Roosevelt immigrant to America. Claes Maartenszen van Rosenvelt arrived in America in 1649, and while the records are spotty, it appears that he came over with wealth, since he had brought over a portrait of himself painted back in the Netherlands, and portrait paintings weren't something middle class or lower class people could well afford at the time. The portrait hangs at the Springwood Estate, the home that FDR grew up in, which is now preserved as the Home of Franklin D. Roosevelt National Historic Site.
Claes's son, Nicholas, made a bunch of money in the fur trading business and served as an alderman in New York City in the early 1700s. His son, Jacobus, made a bunch of money in real estate. He bought swampland that later became known as the neighborhood of Five Points in Lower Manhattan. He developed it into separate lots, and sold them off, and made a fortune in the mid-1700s. He also bought hundreds of acres in New Jersey and made a fortune there, too.
His son, FDR's great-great-grandfather, Isaac Roosevelt, was in the New York State Senate during and after the American Revolution, and became the 2nd President of the Bank of New York, replacing the departing Alexander Hamilton who had become U.S. Treasury Secretary.
FDR's great-grandfather James was also a banker and was elected to the New York State Assembly and served for a time as a councilman on the New York City Council.
FDR's grandfather Isaac was another businessman, and inherited considerable wealth, built a large house on a large estate in upstate New York he named Rosedale, and spent his leisure time competing in yacht races. At one time, he owned the world's largest ice yacht.
FDR's father James was a Harvard-educated lawyer who was the Vice-President of the Delaware and Hudson Railway and the President of the Southern Railway Security Company, and served for a time in the U.S. Embassy in the United Kingdom under James Buchanan (who became president not long after). He bought the Springwood mansion in 1866.
FDR was a well-connected guy whose family wealth went back many, many generations. It is true that his maternal grandfather made money in opium, but that was just one aspect of the wealth of FDR's family. A fortune was made in the opium trade by one of FDR's grandfathers, but it wasn't the fortune, because the family had made multiple fortunes dating back hundreds of years, and FDR benefited from all of them. Had Warren Delano, Jr., never engaged in opium smuggling, it probably wouldn't have much affected his station in life as a very wealthy man. He succeeded in making money through a lot of different means, though opium was certainly one which he profited from considerably.
Details of the generational wealth along FDR's paternal line are probably most succinctly found in Franklin D. Roosevelt: A Biography by Jeffrey W. Coker or The Amazing Roosevelt Family, 1613-1942 by Karl Schriftgiesser. Another alternative is Franklin D. Roosevelt: The Apprenticeship by Frank Freidel, the first volume of a three-volume biography. Other details can be found in The Roosevelt Genealogy, 1649-1902 by Charles Barney Whittelsey.