Well Switzerland didn't for example take the move during the cold war to officially join the non-alignment league that would have made their stance particularly explicit on wishing to avoid participation in the cold war though particularly early on many of the members of said league were economically left leaning groups of various ideological persuasions that wished to distance themselves from he Soviet Union to a certain extent such as Yugoslavia, India and Egypt which is a label which of course couldn't exactly be applied to Switzerland.

It also didn't do anything explicit such as choose to join an official military alliance such as NATO. While it has pursued a partnership with NATO in more recent years such as some cooperation during the Yugoslav Wars with NATO lead peacekeeping efforts in the Balkans and joining the Partnership for Peace in 1996 such efforts were largely not pursued during the cold war period.

It was relatively late it should be noted that Switzerland chose to establish diplomatic relations with the Soviet Union, 1946 arguably in response to the growing tensions that were beginning to building. One of the last countries in Europe that were not new ones to do so. Part of the road towards the normalisation of relations involved things like freeing up previously frozen assets the Soviets government had claim to from Swiss banks. One of the major concessions the Swiss government had to make was that during the second world war around ten thousand Soviet soldiers who'd been taken prisoner by the Germans had, mostly towards the very end of the war, escaped and fled to neutral Switzerland for safety. Some would have preferred to remain in Switzerland a few having been deserters from the red army or where in hot water for political reasons. Switzerland chose to extradite them back to the Soviet Union anyway. I think this demonstrates the lengths which the Swiss government was willing to go to to pursue a neutral position with the Soviet Union.

Switzerland did maintain a strong civil defence system and arguably it has specifically chosen to do a lot of things like building as much redundancy in certain things like water infrastructure as possible. But while it wouldn't have been in Switzerland's interest to see the international economic system become a communist lead one they perhaps did not believe that them joining NATO or any other anti-soviet alliance. The risk of doing so was perhaps considered just too great. Switzerland was unlikely to win the cold war for the west (and it wasn't clear that the cold war could or even should be out right won from many people's point of view), if they west won on its own Switzerland was unlikely to have actually gained much from joining NATO or the like in political terms. Given the geography of the situation it was exceedingly unlikely that Switzerland would have gained the benefit of the mutual defence clause of the treaty in the sense that if Switzerland was attacked by the Soviet Union other countries coming to fight the Warsaw Pact in its defence wouldn't be necessary as the Soviet Union would almost certainly have invaded other NATO members first meaning NATO would already be at war with them. Meanwhile under the circumstances in which he cold war went hot under other circumstances while Switzerland would be in a better position to coordinate any conflict with the Warsaw Pact it might have needed to fight should the Soviets for what ever reason decide to attack Switzerland but it would also lose the chance to avoid becoming involved in such a conflict. Not to go too deep into alt history scenarios here but it seems to me that had the cold war gone hot it would not have been in the Soviet Union's interests to invade Switzerland unless it had already destroyed NATO in Europe entirely already just as it wasn't a priority for the Nazis. Switzerland would be a both relatively costly and pointless nut to crack if you could avoid doing so.

So from Switzerland's perspective its easy to see why they might conclude they had more to loose than they had to gain by joining an organisation like NATO. And thus while they were more economically tied to NATO countries generally they had no reason to take part in any of the alliance building and proxy conflict aspects of the Cold War.

Edit: Actually i think i should also mention that from a constructivist perspective i think you could make a strong argument that much of the power politics and military arguments i've made here could as easily be applied to counties like Portugal and hence you also have to consider the idea of the national self image. The argument from that perspective would be that countries often act as much as anything else to preserve and further how they see themselves as a nation. Another example is a country that sees itself as a great power is less likely to back down from threats or insults because it things great powers shouldn't stand for that kind of thing. An important aspect of that for the Swiss government compared to some others is neutrality is in some ways more than just a decision but by the point of the cold war had arguably begun to become an actual part of swiss idenity they did not want to compromise.