Does “je bois du café” make correct use of partitives?

Yes, ‘Je bois du café‘ correctly uses the partitif and could generally mean ‘I drink coffee‘ or ‘I am drinking coffee‘.

Since you have some doubts about using the partitif and you have not edited your question to make it easier to answer, I will go ahead and explain to you why the partitif is needed here.

Le partitif:

Construction: partitif article + definite article (de + le / la)

Note: ‘de + le’ contracts into ‘du’

When is it needed in French?

The partitive is needed when talking about things that are not countable. Many types of food and beverages fall into this category.

For example:

Tea
Coffee
Water
Beer
Wine
Bread
Salad
And many more

Now you may be thinking: In English, these are all countable! I had two coffees with breakfast, a salad for lunch, a beer with dinner, and a tea before bed!

Technically, yes, in colloquial English we do say these things but technically the above sentence actually implies the following: I had two cups of coffee with breakfast, a bowl of salad for lunch, a bottle of beer with dinner, and a cup of tea before bed.

You may argue that in English, a bowl of salad is not really implied. This may or may not hold true depending on who you talk to and where. It is about colloquialism, and this varies in different regions where English is spoken.

Why can’t I count it?

Imagine trying to count water. You can’t. You can try to measure it in units of volume. You could try to count the number of molecules, or drops, but those are all measurements. Even if you did know how many litres, bottles, molecules, or drops of water there are, you can’t say there are x waters. You could only say there are x bottles, litres, drops, or molecules of water.

Putting it all together:

Now that you understand why the partitif is used, you can start to understand how to use it in French. Failure to use the partitif construction in French when talking about things that are uncountable would end up sounding as weird as saying in English: ‘I would like a bread‘ or ‘I am eating a bread‘.

Thus, ‘Je bois du café‘ is correct! It translates generally to ‘I am drinking some coffee‘.

You aren’t confined to the partitif! When talking about these uncountable things, you can make it countable by adding some sort of measurement.