Stop swiper slide autoplay on mouse enter and start autoplay on mouse leave

You need to use the option disableOnInteraction: true rather than binding the events yourself see here for documentation.

Optionally you can use the following for autoplay start stop

• swiper.autoplay.start();

• swiper.autoplay.stop();

Edit

Your mistake is how you are getting the instance for swiper. see below for demo

$(document).ready(function() {
  new Swiper('.swiper-container', {
    speed: 400,
    spaceBetween: 100,
    autoplay: true,
    disableOnInteraction: true,
  });
  var mySwiper = document.querySelector('.swiper-container').swiper

  $(".swiper-container").mouseenter(function() {
    mySwiper.autoplay.stop();
    console.log('slider stopped');
  });

  $(".swiper-container").mouseleave(function() {
    mySwiper.autoplay.start();
    console.log('slider started again');
  });
});

.swiper-slide {
  text-align: center;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/Swiper/4.0.5/js/swiper.min.js"></script>
<link href="https://cdnjs.cloudflare.com/ajax/libs/Swiper/4.0.5/css/swiper.css" rel="stylesheet" />
<!-- Slider main container -->
<div class="swiper-container">
  <!-- Additional required wrapper -->
  <div class="swiper-wrapper">
    <!-- Slides -->
    <div class="swiper-slide">Slide 1</div>
    <div class="swiper-slide">Slide 2</div>
    <div class="swiper-slide">Slide 3</div>
    <div class="swiper-slide">Slide 4</div>
  </div>
  <!-- If we need pagination -->
  <div class="swiper-pagination"></div>

  <!-- If we need navigation buttons -->
  <div class="swiper-button-prev"></div>
  <div class="swiper-button-next"></div>

  <!-- If we need scrollbar -->
  <div class="swiper-scrollbar"></div>
</div>

var swiper = new Swiper('.swiper-container', { 
....
...
....
});

$(".swiper-container").hover(function() {
    (this).swiper.autoplay.stop();
}, function() {
    (this).swiper.autoplay.start();
});

generic solution for multiple sliders on same page.

For several instances the only code works is the following:

$(".swiper-container").each(function(elem, target){
    var swp = target.swiper;
    $(this).hover(function() {
        swp.autoplay.stop();
    }, function() {
        swp.autoplay.start();
    });
});

New parameter has been added to Swiper v6.6.0 (pauseOnMouseEnter), so there is no need for a workaround from now: https://swiperjs.com/swiper-api#autoplay

According to the documentation of swiper.js version 8 https://swiperjs.com/swiper-api#autoplay

autoplay: {
    disableOnInteraction: false,
    pauseOnMouseEnter: true,
},

In vuejs (with vue-awesome-swiper), I had to wrap the swiper component with a div and then added the event listeners to the div.

<v-flex @mouseenter="$refs.swiperRef.swiper.autoplay.stop()"
        @mouseleave="$refs.swiperRef.swiper.autoplay.start()">
    <swiper :options="swiperOptions"
            ref="swiperRef">
        <swiper-slide v-for="(product, index) in products"
                      :key="index">
            <!-- -->
        </swiper-slide>
    </swiper>
</v-flex>

Putting the event listeners on the component directly did not work

For Vue 3 with vue components from swiper/vue:

<template>
  <swiper
      :slides-per-view="1"
      @swiper="onSwiper"
      @mouseenter="() => this.swiper.autoplay.stop()"
      @mouseleave="() => this.swiper.autoplay.start()"
  >
    <template v-for="pic in pictures">
      <swiper-slide>
        <figure>
          <div v-html="pic.html"></div>
          <figcaption>{{ pic.title }}</figcaption>
        </figure>
      </swiper-slide>
    </template>
  </swiper>
</template>

<script>
// Import Swiper Vue.js components
import { Swiper, SwiperSlide } from 'swiper/vue';

// Import Swiper styles
import 'swiper/swiper.scss';

export default {
  components: {
    Swiper,
    SwiperSlide,
  },
  data() {
    return {
      swiper: null,
    }
  },
  methods: {
    onSwiper(swiper) {
      this.swiper = swiper;
    },
  },
}
</script>

In case you are using vuejs directive for swipper : vue-awesome-swiper

<div class="swiper-container" v-swiper:mySwiper="swiperOptions" @mouseenter="stopSwip($event)" @mouseleave="startSwip($event)">
    ....
</div>


<script>
export default {
  methods: {
    stopSwip(event) {
        event.target.swiper.autoplay.stop();
    },
    startSwip(event) {
        event.target.swiper.autoplay.start();
    },
  },
}
</script>