Fort Luxemburg

It is possible to survive alone as Luxemburg by building 10 land forts and just sitting there. It will be one of the most boring games you are ever likely to play though and you will not be influencing anything at all until well after the war for Europe has been won (and even then you can only ally with the victor or you will still be unable to do anything). Far better to ally with one side or the other early on.

Find an Ally

First of all, make sure to increase your interventionism each year, as you need either a high interventionism or a dictatorship government to enter alliances while still at peace. Answer events to gain yourself more interventionism and authoritarianism: when "Law Policy Initiative" comes up, choose to "Side with the Ultrarightists", and when a "Rogue Politician" appears, certainly "Ignore him". With the right events, you should have a Paternal Autocrat government by 1937.

Joining the Allies

Allying with the allies while having these 10 forts will not noticeably affect their chances in Europe as there is still plenty of space north of you for the axis to roll through.

Joining the Axis

You could try getting an alliance with Germany so that they conquer France through your territory, giving you French provinces. This will not work with some of the patches, unfortunately (1.2 and up?), as Germany will not take provinces for you just by passing through your country.

To obtain an alliance with Germany, start from the 1936 campaign and immediately begin trading your metal and a tiny bit of rare materials with Germany for enough energy to stay positive and as much money as you can get. If necessary, forgo both research and production in order to produce more money. This trade with Germany will gradually improve your negative relations; the more favourable the trade to Germany, the more quickly relations will improve. As you start gaining energy from trade, you will automatically convert some of this to a tiny amount of oil. Give this oil to Germany in a separate trade agreement.

When your relations with Germany reach +50, this is the highest the trade agreements will raise you. At least by the beginning of 1938, start influencing Germany. By this time, your cash reserve and your interventionism level will be so high that you should have enough money to get German relations high enough so that the "Join Alliance" option will have 100% chance of success. When this succeeds, you can end all one-sided trade agreements and maybe actually begin some research or production.

Once you are part of the Axis, your main worry is France. If Germany and the Allies go to war, France will invade you immediately. Either take military control of Germany to reinforce Luxemburg heavily, or, if you don't like taking military control, make certain that Germany never goes to war with the Allies until you are ready to defend Luxemburg. One way of doing this is by declaring war on Tibet, or anyone who isn't in danger of overrunning you. This should stop the "Danzig or war" event from firing, and it will hopefully guarantee a war that will not reach your tiny duchy.