The question "Is there a basic particle?" seems to me to mean "Is there some most fundamental constituent of matter?" Such particles are not unique- there are in fact twelve of them- but one the the triumphs of modern science in the Standard Model of particle physics. That model is, hand down, THE most thoroughly and most precisely verified physical model of all time. Every since the Steven Weinberg and Abdus Salam proposed the Standard Model, more than three decades ago, experiments have been being run continuously all over the world first to verify the Standard Model and then to look for its limitations. To date, no single instance has ever occurred inconsistent with the Standard Model. As a scientist, I can tell you that this situation is frustrating, contrary to the popular conception of scientists. The short version however is that for technical theoretical reasons, we believe the Standard Model is an effective field theory, an approximation to some more precise model. Without finding out where the Standard Model goes wrong, it's difficult to imagine how we can get a hint even at a non-effective or non-approximate theory.
So what exactly does this perhaps too successful Standard Model involve? Mathematically, it describes all physical interactions apart from gravity in terms of group theory, a branch of mathematics which describes similarities of different situations. Nevertheless, the Standard Model at the same time describes two classes of particles which make up matter ["quarks" and "leptons"] and a third class ["Bosons"] via which those particles interact. Bosons include four particles which have been observed [the photon of which light consists and the less familiar W- which can be electrically positive, negative or neutral-, Z- which is electrically neutral- and gluon- via which quarks interact and which has "color" charge] and possibly a graviton via which some hypothesize that bodies interact gravitationally but which has yet to be observed. Leptons consistent of the electron- the interactions of which in the atom explain chemistry-, the muon and the tau, along with a related species of neutrino for each of these. Quarks are peculiar objects which interact via a "color" charge- a term which started as a physicists' joke but stuck- but occur only in combinations where the color charge cancels. Familiar protons and neutrons are made of combinations of quarks called up and down quarks. More exotic objects may contain strange or charm quarks. Even more rare object contain quarks referred to as top and bottom by the American physics community and truth and beauty by the European physics community. [Those wishing to be neutral just refer to them as t and b quarks.] Everything in the universe we know of consists of these particles and only these particles in various combinations.
The question then arises how we know these particles do not themselves consist of yet smaller finer particles. String theorists argue that perhaps these particles do in fact consist of finer particles far too small to detect, but for now at least no experimental evidence of this exists. One of the reasons physicists use particle colliders is that particles with an internal structure- those consisting of smaller particles- behave differently in a collision than do particles without any internal structure, i.e., "basic" particles in the terminology of the debate. Protons and neutrons do in fact display signs of having an internal structure and this is part of the evidence for the existence of quarks, which to date have not themselves shown any evidence of having an internal structure. Leptons likewise do not show evidence of having an internal structure. So, while it is possible that new evidence will come to light, to date- and as I stated at the outset the Standard Model is overwhelmingly the most throughly and precisely tested physical model ever- all evidence indicates that a small set of "basic particles" does exist: six quarks, six leptons and four [maybe five] bosons.