登录时记住密码功能一般可以用sessionStorage，sessionStorage,Cookie实现，以下表格是三者间的差异：

图片来源
此处我使用的是Cookie来实现记住密码功能（js-cookie）,使用jsencrypt对密码进行加密解密
npm i js-cookie -s
npm i jsencrypt -s
实现完整代码：

<template>
  <el-form
    ref="ruleFormRef"
    :model="ruleForm"
    status-icon
    :rules="rules"
    label-width="80px"
    label-position="left"
    class="demo-ruleForm"
  >
    <el-form-item label="用户名" prop="username">
      <el-input v-model="ruleForm.username" />
    </el-form-item>
    <el-form-item label="密码" prop="pass">
      <el-input v-model="ruleForm.pass" type="password" autocomplete="off" />
    </el-form-item>
    <el-form-item label="" prop="remerber">
      <el-checkbox
        v-model="ruleForm.remerber"
        label="记住密码"
        size="large"
        style="float: left"
      />
    </el-form-item>
    <el-form-item>
      <el-button type="primary" @click="submitForm(ruleFormRef)"
        >登录</el-button
      >
      <el-button @click="resetForm(ruleFormRef)">重置</el-button>
    </el-form-item>
  </el-form>
</template>

<script lang="ts" setup>
import { reactive, ref } from "vue";
import { CommonPicker, type FormInstance, type FormRules } from "element-plus";
import Cookies from "js-cookie";
import { encrypt, decrypt } from "./jsencrypt";

const ruleFormRef = ref<FormInstance>();
const ruleForm = reactive({
  username: "",
  pass: "",
  remerber: false,
});

const rules = {
  username: [{ required: true, trigger: "blur", message: "请输入您的账号" }],
  pass: [{ required: true, trigger: "blur", message: "请输入您的密码" }],
};

const submitForm = (formEl: FormInstance | undefined) => {
  if (!formEl) return;
  formEl.validate((valid) => {
    if (valid) {
      if (ruleForm.remerber) {
        Cookies.set("username", ruleForm.username, { expires: 30 }); //设置有效期为30天
        Cookies.set("password", encrypt(ruleForm.pass), { expires: 30 });
      }else{
        Cookies.remove('username')
        Cookies.remove('password')
      }
    } else {
      return false;
    }
  });
};

const resetForm = (formEl: FormInstance | undefined) => {
  if (!formEl) return;
  formEl.resetFields();
};

const getCookie = () => {
    let username = Cookies.get('username')
    let pass = Cookies.get('password')
    ruleForm.username = username === undefined?ruleForm.username:username
    ruleForm.pass = pass === undefined?ruleForm.pass:decrypt(pass)
};
getCookie()
</script>
<style scoped lang="less">
.demo-ruleForm {
  position: absolute;
  left: 50%;
  top: 50%;
  transform: translate(-50%, -50%);
  width: 400px;
  background-color: antiquewhite;
  padding: 40px 20px;
  border-radius: 10px;
}
</style>

jsencrypt.ts：
密钥对生成 http://web.chacuo.net/netrsakeypair

import JSEncrypt from 'jsencrypt/bin/jsencrypt.min'
const publicKey = ``
const privateKey = ``

// 加密
export function encrypt(txt:string) {
  const encryptor = new JSEncrypt()
  encryptor.setPublicKey(publicKey) // 设置公钥
  return encryptor.encrypt(txt) // 对数据进行加密
}

// 解密
export function decrypt(txt:string) {
  const encryptor = new JSEncrypt()
  encryptor.setPrivateKey(privateKey) // 设置私钥
  return encryptor.decrypt(txt) // 对数据进行解密
}

页面：

控制台可看到储存的cookie