In Chulin 7b it says: "a person does not stub his toe unless it was decreed upon him on high." How is it known that this, and hashgacha pratis in general, is still applicable today?

As opposed to Jewish history (where G-dly direction is unquestionable in the modern world and especially in Israel and the ingathering of the exiles, the wars, etc.), there is a difference of opinion regarding the extent of Divine intervention for individuals. The Chassidic masters cite the gmara which you mentioned and take it further, even applying it regarding the animal world. They hold that sometimes G-d will make an entire storm in order to move one blade of grass closer to one worm! Contrarily, the Rambam writes that G-d intervenes and over-watches the details only by the deserving, and leaves the “masses”, the animals and the vegetation to be governed by the laws of nature which He created for that purpose (Guide for the Perplexed, p. 315). Accordingly, the Rambam would explain the passage you cited as referring to the deserving ones. Over the years, I have seen enough clear examples to lean in the direction of the first approach, surely regarding Jewish (!) individuals. I suggest you ask around, and you will probably hear of many such personal cases of hashgacha on people, from those close to you, and from many non-Jews, as well.