Was the danger of Germany bombing the US East Coast taken seriously and were measures taken to address it?

Short Answer:
Very seriously. on both coasts. In retrospect more serious than was probable reasonable. A US air and sea defensive system was brought online beginning in 1940 modeled after the British front line air and sea programs which at the time were winning the Battle of Britain. Their were millions involved. Plane spotters and air wardens and coastal defenses built on both coasts. But beyond the coasts the program was active hundreds of miles inland and around the country. Submarines were an especially big issue on the east coast early on. German submarines patrolled and took merchant shipping within eyesite of America's largest cities.
The War That Never Came: Civilian Defense in Cincinnati, Ohio During World War II
During World War II the Office of Civilian
Defense (OCD), a federal agency, encouraged more than ten
million Americans to volunteer their time and effort for all
types of defense-related activities that provided psychological
and material benefits for the home front. The agency's top
priority was civilian protection. By the summer of 1941, air
raid precautions modeled after Great Britain's during the blitz had been developed in the United States. Treated solemnly
by many, especially in the aftermath of Pearl Harbor, civilian protection was seen as America's last line of defense.

Detailed Answer:
In the 1960's I toured a series of watch tower fortifications on the DelMarva Peninsula. They were basically concrete bunkers with a tower for coastal watching. These were built all up and down the east coast. On the coast both planes and and ships were aggressively watched for. They also looked for spies and sabatours landings on the beaches. Caught a few too.
7 crazy things the Coast Guard did during World War II
In June, 1942, a German U-boat surfaced off the coast of New York and dropped off a team of four saboteurs that made their way to the coast. Their goal was to cripple U.S. aluminum production and hydroelectric power production through a terror campaign, weakening the U.S. and hopefully coercing the U.S. population to vote against the war. The endeavor was quickly foiled thanks to the Coast Guard beach patrol.
and
German saboteurs executed in Washington, D.C.
I remember a German u-boat commander Hardegen, who sank about 24 ships in American waters on two patrols in 1942. The most famous one which always sticks with me was one I heard about as a kid, many years after the fact. He sank a ship right off the coast of Coney Island, New York and people on the Ferris wheel got a good view of his Uboat. Commander Hardegen was a very professional fellow. He had hit the merchant man with a torpedo and it was on fire but still afloat. Commander Hardegen then surfaced to finish the ship off with his deck gun. Only he decided to circle the ship and come in from the shore side so his deck gun would not accidently take out people at the amusement park. ( we know this because he was interviewed after the war. ) Only when he did so his uboat was illuminated to the horrified people on shore could see his ship go in for the coup de grace. It was a kind of game changer at the home front. That really struck home with the people of the US.
U-boat commander who menaced American Shores
I also have some pictures of my Grandfather who was an air warden in Cincinnati Ohio. He had an armband and a WWI style helmet. When I questioned him about it, as a kid. He said they did things like plane spotting and walked the streets during black outs to make sure people turned off their lights or if in a car pulled over and turned off their headlights. Cincinnati is about 600 miles from the coast.
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WWII documents reveal importance of air raid wardens
One more story for you. Washington DC is about 200 miles from the coast. During WW2 there were anti aircraft guns placed around important American Cities including Washington DC. One of these sites on the Capital Mall right near the federal center (legislature, and whitehouse). The soldiers manning there gun fired it. Not sure if it was because they misidentified a plane or “the battery went off by itself”. The net effect was they hit/shelled the Lincoln memorial. The us park service at the memorial will proudly state that the Lincoln Memorial is the only Federal Memorial ever to come under fire during war time. Always thought that their pride kinda came at the expense of the US army.
Washington During Wartime
One overeager air defense soldier accidentally machine gunned the Lincoln Memorial in 1942 (seriously, this actually happened). Unfortunately, the only contemporary news coverage of this incredible event was a brief Washington Post story, and an even shorter NYT blurb.