First, I would argue that it is fallacious to say that the US was every truly "isolationist". Ever since independence, the US tried to and necessarily had to maintain relations with the other nations of the world. A better term, in my opinion, would be "unilateralist" as the US often acted on its own accord rather than within the confines of Old World diplomacy dominated by European powers and issues.

Second, so-called "isolationist" policies like the Smoot-Hawley Tariff definitely helped to exacerbate the Great Depression. However, it really depends on who you ask for what caused the Great Depression and why it lasted so long. A nice overview of the aggregate reasons are that rising interest rates prior to the stock market crash led to a cash crunch, which helped to spur economic pessimism in the country that led to people limiting consumption and investment, which resulted in a depressed aggregate demand. The inability of the government to contend with people's sinking expectations of the economy was what really helped exacerbate the issues and prolong the Depression as people would have no reason to spend or invest if they think things are just going to get worse. There are also many other issues that led and prolonged the Depression.

"Isolationist" policies like S-H definitely exacerbated the Depression, but it would be short-sighted to put too much weight on them individually.

This question seems to be US-focused, but similar policies were followed in many countries. The 1930s is not a period I ever specifically studied, but with the help of some background knowledge and internet research I can hopefully shed some light.

Trade policies in the 1930s are characterised as "beggar-thy-neighbour".

Here's roughly how it worked. A country saw that 1) it has a trade deficit with another country and was building up a debt, or 2) cheaper imports were undercutting local products, meaning firms were laying people off, closing, etc.. It's worth noting that the gold standard meant that the value of the currency could not depreciate to help balance imports & exports.

So, the government decides to introduce trade barriers. Typically these were tariffs on imports but they could also be quotas or other measures. The idea of tariffs is that if imported goods cost more, people (and industries) will buy less. Fewer imported goods reduces the trade deficit, and means people will buy more locally produced goods, stimulating the national economy. Great!

Except that other countries now see imports from your country undercutting their products, and start getting upset if they have a trade deficit with you. So they start putting up barriers against your goods.

The end result is that there are barriers to trade all over the place and trade falls.

Now, a fundamental assumption of liberal economics is that trade is a good thing. It allows competitive advantage to flourish. If I can produce oranges more efficiently than you and you can produce hats more efficiently than me, it benefits both of us to trade. It doesn't matter if that trade is between towns or between countries.

However that view can and has been criticised, particularly by development economists. Some form of protection can help "infant" industries to grow (or give declining industries time to recover) without being competed out by larger, more developed international competitors. Most developed countries have been through some form of protectionist regime. The risk is that because they're isolated from competition, industries become or remain inefficient, which either hurts your own consumers or means they'll be out-competed when barriers are relaxed. The trick seems to be getting the policies right, and relaxing them at the right time, and it's not entirely clear whether that can be done by judgement more than luck.

To look at a concrete example: the UK. Bear in mind all of this is specific to the UK and the experience of other countries - perhaps the US especially - may be very different.

The UK began raising tariffs in 1931 with the Emergency Abnormal Importations Act. This was followed by the Import Duties Act in 1932, which imposed a 10% tariff on manufactured imports (with various exceptions). Duties were raised further over the course of the next few years.

Increasing the price of something doesn't necessarily reduce consumption, so did these duties actually decrease imports? The 'beggar they neighbour' paper linked below finds that increased prices actually didn't reduce imports, but imports did nonetheless fall. The authors put this down to businesses and consumers responding to calls to import less.

What was the impact of these policies on the economy as a whole?

That's less clear. For the UK at least, it doesn't seem to have had a major impact on growth. There are arguments that reducing industrial imports actually had a beneficial effect. There were effects on productivity, because industries were sheltered from competition. However whether this was worth it if it protected employment is a trickier question, and partly a normative one. (What's more important, jobs now, or higher productivity, lower prices and maybe more jobs later?)

As an endnote, the tariffs introduced in the UK in the early 30s were retained through the 40s and 50s, and have taken some blame for the low productivity of British industry in the post-war period. As I said earlier, the longer-term effects of barriers can be negative, if they're not removed at the right time.

Sources:
Returning to Growth: Lessons from the 1930s
Beggar Thy Neighbour: British Imports During the Inter-War Years and the Effect of the 1932 Tariff
The macroeconomics of protectionism: the case of Britain in the 1930s

I actually just answered a bit about Smoot-Hawley here.